Nebraska Open Access Mathematics - User contributions [en]

4.6 Algebraic Fractions

2020-07-06T15:15:51Z

Nwakefield2: /* Suggested Lecture Notes: */

=Lesson Plan 4.6: Algebraic Fractions=

==Objectives:==

* Understand how to simplify algebraic fractions by factoring and employing algebraic manipulations.

==Suggested Lecture Breaks:==
*'''MWF''': You have two days. Split the material where it feels natural to you.
*'''MW/TR''': You have one day.

==Suggested Lecture Notes:==

* Begin the class by asking students what an ``algebraic fraction" is, which should be familiar from their RG. Then put up the following definition on the board:
** An ''algebraic fraction'' is the quotient of two polynomials, where the denominator is not zero.
* Work through an example of simplifying an algebraic fraction. One example might be:
**<math>\dfrac{x^5(x^2-4)}{x^2(x+2)(x+3)}</math>

*Point out that whenever we cancel common factors we must place assumptions on the value(s) of the variable(s). To this end, define restriction:
** A ''restriction'' is a number for which an expression is undefined.

* Work through some examples of adding/subtracting algebraic fractions. Be sure at least one of these examples requires simplifying an algebraic fraction that involves factoring a quadratic in the denominator. Here are some suggested examples:
** <math>\dfrac{1}{5} + \dfrac{1}{y}</math>
** <math>\dfrac{2}{3x^2} - \dfrac{3}{7x}</math>
** <math>\dfrac{x+1}{3x+7} + \dfrac{5x-2}{4}</math>
** <math>\dfrac{x^2-2x+3}{x^2+7x+12} - \dfrac{x^2-4x-5}{x^2+7x+12}</math>
** <math>\dfrac{4x}{x^2+x-12} - \dfrac{3}{x^2-9}</math>
** <math>\dfrac{6}{x-2} + \dfrac{x+3}{2-x}</math>
* Next, work through a couple examples of simplifying algebraic expressions involving multiplication and/or division. Again, emphasize that we must place assumptions on the value(s) of the variable(s) when we cancel common factors. Often we will simply place a statement in the instructions such as, ``Assume any factors you cancel are not zero" to address this issue. Here are some suggested examples:
** <math>\dfrac{5p}{6q^2}\cdot\dfrac{3pq}{5p}</math>
** <math>\dfrac{3y^4}{4z}\cdot\dfrac{8y^3z}{6y^5}</math>
** <math>\dfrac{x+3}{x+4}\div\dfrac{4x+12}{2x+8}</math>

==Notes==

* As you work through the examples, explain why we must find a common denominator and how we use factoring to help us in this process.
* Reinforce that using the least common denominator will result in less of a need to simplify in the end, but any common denominator will do.
* Many times students want to multiply out the common denominator; please discourage this by pointing out that it just causes more work later.
* Be sure you relate this topic back to the previous lessons on factoring quadratic expressions and the lessons from the first week involving equivalent fractions and algebraic manipulation of fractions.

==Comments on the handout:==

*'''Questions 1 & 2:''' These questions should be routine for students. If you find that you are running out of time, you can ask students to do questions 1 and 2 at home. Otherwise, you should allow students to work in class on these first two question a maximum of 10 minutes.
* Because this is the last section that will be tested on the final, you might want to offer detailed solutions to students or plan on spending some time on it at the beginning of the next class period, if time allows.

4.6 Algebraic Fractions

2020-07-06T15:14:49Z

Nwakefield2: /* Suggested Lecture Notes: */

=Lesson Plan 4.6: Algebraic Fractions=

==Objectives:==

* Understand how to simplify algebraic fractions by factoring and employing algebraic manipulations.

==Suggested Lecture Breaks:==
*'''MWF''': You have two days. Split the material where it feels natural to you.
*'''MW/TR''': You have one day.

==Suggested Lecture Notes:==

* Begin the class by asking students what an ``algebraic fraction" is, which should be familiar from their RG. Then put up the following definition on the board:
** An ''algebraic fraction'' is the quotient of two polynomials, where the denominator is not zero.
* Work through an example of simplifying an algebraic fraction. One example might be:
**<math>\dfrac{x^5(x^2-4)}{x^2(x+2)(x+3)}</math>

*Point out that whenever we cancel common factors we must place assumptions on the value(s) of the variable(s). To this end, define restriction:
** A ''restriction'' is a number for which an expression is undefined.

* Work through some examples of adding/subtracting algebraic fractions. Be sure at least one of these examples requires simplifying an algebraic fraction that involves factoring a quadratic in the denominator. Here are some suggested examples:
** $\dfrac{1}{5} + \dfrac{1}{y}$
** $\dfrac{2}{3x^2} - \dfrac{3}{7x}$
** $\dfrac{x+1}{3x+7} + \dfrac{5x-2}{4}$
** $\dfrac{x^2-2x+3}{x^2+7x+12} - \dfrac{x^2-4x-5}{x^2+7x+12}$
** $\dfrac{4x}{x^2+x-12} - \dfrac{3}{x^2-9}$
** $\dfrac{6}{x-2} + \dfrac{x+3}{2-x}$
* Next, work through a couple examples of simplifying algebraic expressions involving multiplication and/or division. Again, emphasize that we must place assumptions on the value(s) of the variable(s) when we cancel common factors. Often we will simply place a statement in the instructions such as, ``Assume any factors you cancel are not zero" to address this issue. Here are some suggested examples:
** $\dfrac{5p}{6q^2}\cdot\dfrac{3pq}{5p}$
** $\dfrac{3y^4}{4z}\cdot\dfrac{8y^3z}{6y^5}$
** $\dfrac{x+3}{x+4}\div\dfrac{4x+12}{2x+8}$

==Notes==

* As you work through the examples, explain why we must find a common denominator and how we use factoring to help us in this process.
* Reinforce that using the least common denominator will result in less of a need to simplify in the end, but any common denominator will do.
* Many times students want to multiply out the common denominator; please discourage this by pointing out that it just causes more work later.
* Be sure you relate this topic back to the previous lessons on factoring quadratic expressions and the lessons from the first week involving equivalent fractions and algebraic manipulation of fractions.

==Comments on the handout:==

*'''Questions 1 & 2:''' These questions should be routine for students. If you find that you are running out of time, you can ask students to do questions 1 and 2 at home. Otherwise, you should allow students to work in class on these first two question a maximum of 10 minutes.
* Because this is the last section that will be tested on the final, you might want to offer detailed solutions to students or plan on spending some time on it at the beginning of the next class period, if time allows.

7.2: Power Functions

2020-06-01T14:46:21Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Determine whether a given function is a power function by rewriting..."

[[7.1: Combining Functions | Prior Lesson]] | [[8.1: Polynomial Functions | Next Lesson]]
==Objectives:==
* Determine whether a given function is a power function by rewriting it in the form $f(x) = kx^p$
* Classify power functions and their graphs into four basic types
* Describe the end behavior of power functions

==Important Items==
===Definitions:===
power function

===Notes:===
This lesson should prepare students to learn about polynomials and long-run behavior of rational functions. We will not work with power functions with fractional exponents $p$ or discuss direct or inverse proportionality.

==Resources==
[[Media:Section 11-2 ClassificationTable.pdf|Here]] is a review sheet for the four classifications of graphs of power functions put together by a 101 instructor. You can find the tex file for this in Box in the Resources folder.

==Lesson Guide==

===Warm-Up===

Have students do Problem 1.

===Determine whether a given function is a power function by rewriting it in the form $f(x) = kx^p$===

A [[power function]] is a function that can be written in the form $f(x) = kx^p$, where $k$ and $p$ are any constants.

Give several examples and non-examples of power functions. Give an example of a power function that is not yet in the form $y=kx^p,$ and demonstrate how to manipulate the function into this form to prove that it is indeed a power function.

*Example:
The area of a circle is a function of the radius: $A(r) = \pi r^2$. To see that this is a power function, note that $k=\pi$ and $p=2$.

* Example:
 
 
 
 
 

* Example:

 
 
 
 
 

Point out that $p$ can be negative, $k$ can involve a number like $e$ or be a fraction, and $x$ cannot be in an exponent.

Have students do Problems 2 and 3.

===Classify power functions and their graphs into four basic types===

Summarize the four types of power functions as follows. For now, consider functions with $k=1$ to focus on the effect of $p$, where $p$ is an integer.

\begin{tabular}{|c|c|c|}
\hline
& $p$ Even & $p$ Odd \hline
$p>0$ & &
& \includegraphics{images/section11_1graph1.png} & \includegraphics{images/section11_1graph2.png}
& e.g. $y = x^2$, $y= x^4$ & e.g. $y = x^3$, $y=x^5$ [1ex] \hline
$p<0$ & &
& \includegraphics{images/section11_1graph3.png} & \includegraphics{images/section11_1graph4.png}
& e.g. $y = x^{-2} = \frac{1}{x^2}$, $y=\frac{1}{x^4}$ & e.g. $y = x^{-1} = \frac{1}{x}$, $y = \frac{1}{x^3}$ [1ex] \hline
\end{tabular}

To extend the above examples of even and odd functions to power functions with leading coefficient different than $k=1,$ remind students of how the constant $k$ affects a power function using what they know from Chapter 5.

Have students do Problem 4 and 5.

===Describe the end behavior of power functions===

Do several examples to introduce end behavior using the notation in Problem 6.

* Example:
 
 
 
 
 
* Example:
 
 
 
 
 

Have students do Problem 6 and 7. Note that Problem 7 might seem challenging to them at first.

7.1: Combining Functions

2020-06-01T14:46:08Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Find formulas for combinations of functions ==Important Items== =..."

[[6.2: The Vertex of a Parabola | Prior Lesson]] | [[7.2: Power Functions | Next Lesson]]
==Objectives:==
*Find formulas for combinations of functions

==Important Items==
===Definitions:===
===Notes:===
The purpose of Chapter 11, "Combining Functions" and "Power Functions", is to prepare students for 12.1 "Polynomials". Specifically, understanding that polynomials are a combination of power functions will help students to understand why we only examine the highest degree term of a polynomial when finding its end behavior.

The purpose of 11.1 is for students to 1) recognize that they have been combining functions all along and 2) develop their word problem solving skills, while applying their knowledge of function transformations and quadratic functions.

Be prepared to spend most of the lesson on group work.
----

==Lesson Guide==

===Warm-Up===
Have students do Problem 1.

Emphasize the importance of using parentheses. You may also need to go over how to use factoring to simplify in part (f).

===Find formulas for combinations of functions===

Students have seen a cost, revenue, profit word problem before in 5.3 "Linear Functions" and also have lots of experience with building simple linear formulas, so you may not need to do an example for your students. However, if you think that they need a refresher, you may want to address finding profit based on a revenue and a cost function.

Optional: Do a profit example, where you build separate functions for cost and revenue.

* Example:
 
 
 
 
 

Have students do Problems 2-4. If you choose not to do a profit example, then it is worth mentioning when discussing Problem 2 with your students that $P(h) = 50h$, because more generally, $P(h) = R(h) - C(h) = 250h - 200h = 50h$.

Since students tend to struggle with Problem 5, it may be helpful to do another example like Problem 5 where the combination of functions is done via multiplication and show students how considering the units of the combined functions it may help to interpret it in real world terms.

* Example:
You own a restaurant and have 5 waiters/waitresses whom you'll pay $15/hour. However, if you are understaffed any given night, you'll pay $1.25/hour extra for each missing waiter/waitress. Write a formula $f(x)$ which gives the number of waiters/waitresses working on an evening where $x$ waiters/waitresses are missing. Write a formula $g(x)$ which gives the hourly wages per waiter/waitress working on an evening where $x$ waiters/waitresses are missing.

$f(x) = 5-x$
$g(x) = 15+1.25x$

What does $f(x)g(x)$ represent? Emphasize that the meaning is in the units.

$f(x)g(x) = (5-x)(15+1.25x)$. If students are taking Chemistry, tell them that it's helpful to use dimensional analysis. We have (waiters)(wages/waiter) = wages, so $f(x)g(x)$ is how much money you need to pay all your waiters/waitresses on an evening where $x$ waiters/waitresses are missing.

Have students do Problems 5 and 6.

6.2: The Vertex of a Parabola

2020-06-01T14:45:58Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Learn about the vertex form of a quadratic function..."

[[6.1: Introduction to Quadratic Functions | Prior Lesson]] | [[7.1: Combining Functions | Next Lesson]]
==Objectives:==
* Learn about the vertex form of a quadratic function
* Find a formula given the vertex and another point on a parabola
* Complete the square

==Important Items==
===Definitions:===
vertex, vertex form, axis of symmetry

===Notes:===

Expect students to be much less familiar with vertex form and completing the square than they were with standard form, factoring, and the quadratic formula. You will need to review completing the square with them.

----

==Lesson Guide==

===Warm-Up===

Have students do Problems 1 and 2.

===Learn about the vertex form of a quadratic function===

Review standard form and factored form from \S6.1. Discuss what information is highlighted in each form, i.e., in standard form we can clearly see the $y$-intercept and in factored form we can clearly see the $x$-intercepts. Introduce vertex form by sketching the graph of a parabola (maybe even use the graph from the warm-up Problem), and ask what important geometric features the graph has. In addition to the intercepts and the concavity (though they probably do not know this term, they might comment on the ``bowl" shape of each graph), say that an important feature of the parabola is the vertex. Point out the vertex in each example. Say the vertex is either the highest point or the lowest point on the graph of a quadratic function, depending on whether the function is concave up or concave down. Just like the standard form highlights the $y$-intercept and the factored form highlights the $x$-intercepts, we have a third way to write the equation for a quadratic function that highlights the vertex.

* Example:
 
 
 
 
 
* Example:

 
 
 
 
 

You may find it valuable to take this opportunity to connect the vertex form to what students learned in chapter 5. Have your groups find the explicit equation of the function whose graph is the graph of $f(x)=x^2$ translated left/right and up/down by some non-zero amount. While they do this, put up a graph of the resulting function. Have them identify the coordinates of the vertex.
The [[vertex form]] of a quadratic function is
\[
y = a(x-h)^2 +k,
\]
where $a$ is a non-zero constant and $(h,k)$ is the vertex of the parabola. We say that the vertical line which is the graph of $x=h$ is the [[axis of symmetry]] of the parabola.

Have students do Problems 3 and 4.

===Find a formula given the vertex and another point on a parabola===

Discuss student solutions to Problem 4, and then produce a graph of the function in Problem 4. Use this graph to talk about how you could also take a graph and derive a formula.

Have students do Problem 5.

===Complete the square===

'''Note:''' If students need or want more help with completing the square, refer them to the OER.

Since completing the square is an algorithm, emphasize to students that the purpose is to convert a quadratic function in one form to vertex form. Do several examples and give thorough steps on how to complete the square.

* Example: \vspace{1.5in}
* Example: \vspace{1.5in}

Have students do Problems 6-7.

Quadratic functions appear in many places in the real world, highlight this again to your students. Choose at least one of problems 8, 9 and Focus problem for students to work through. [https://www.desmos.com/calculator/hckzcwsbyq Here] is a '''cool''' desmos animation for problem 8, if you do it. The demo is also embedded below. If you run out of time to complete the other questions encourage students to finish the last two problems at home or in the MRC.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/hckzcwsbyq?embed
|width=500
|height=500
|border=0
}}

===Comments===
For the pedagogy project (or your own interest), and edited version of this lesson plan can be found here:
http://www.math.unl.edu/~nwakefield2/FYM/index.php/6.2:_The_Vertex_of_a_Parabola/LailaAwadalla

6.1: Introduction to Quadratic Functions

2020-06-01T14:45:47Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Recognize quadratic functions in standard and factored f..."

[[5.6: Combining Transformations | Prior Lesson]] | [[6.2: The Vertex of a Parabola | Next Lesson]]

==Objectives:==
* Recognize quadratic functions in standard and factored forms
* Find $x$-intercepts of quadratic functions by using factors

==Important Items==
===Definitions:===

standard form, parabola, quadratic

Notes: Students have seen quadratic functions, factoring, and the quadratic formula before, but most likely have not mastered these topics. This means that you should rely on the students to generate the basic ideas but add your insights and understanding to the students previous knowledge. It is a mistake to assume that nobody has seen this material before and pretend this is a first presentation. We are trying to help the students understand their previous knowledge. Throughout this chapter, really try to emphasize the different forms of a quadratic function and why some forms are better for answering various questions than the other forms.

----

==Lesson Guide==

===Warm-Up===
Have students do Problems 1 and 2.

===Introduce quadratic functions and standard form===

Introduce the shape of quadratic functions (parabolas) using several real world examples. These can vary widely and can include things like water fountains, the graph of the height of a ball that is thrown straight up into the air, $y=x^2$ graphed on axes, some parabolas concave up/down, and some with 0, 1, 2 $x$-intercepts. Say this shape is called a {\em parabola}, and that functions whose graphs have this shape are called {\em quadratic functions}.

*Example:
 
 
 
 
 

Parabolas show up in real life often because of the effects of gravity (like we see in Angry Birds and the water fountain). If a ball is thrown upward from the top of a building, even if it just goes straight up and down, then its vertical height after $t$ seconds is given by a quadratic function, for example, $h(t) = -16t^2 + 32t + 128$. By the end of class today, we'll be able to find the height from which the ball was thrown and the time when the ball reaches the ground. After tomorrow's class, we'll be able to find the maximum height the ball reaches.

* Example:

 
 
 
 
 

The [[standard form]] for a quadratic function is $y = ax^2 + bx + c$, where $a, b$, and $c$ are constants and $a \neq 0$.
The graph of a quadratic function is called a [[parabola]].

Ask students to recall from Chapter 1 how we find the $y$-intercept of a function. (In general, to find the $y$-intercept of a function $f(x)$, we evaluate $f(0)$.)

Have students recall Problem 1 on the worksheet. Ask students to see if they note a relationship between the standard form of a quadratic function and the $y$-value of the $y$-intercept.

Point out that we can read the $y$-intercept right off the standard form. Ask, ``What does \textbf{c} in standard form tell us about the graph of the quadratic function $f(x) = ax^2 + bx + c$?'' It's the $y$-intercept.

Also note that the function has '''two''' $x$-intercepts. Remind students to be careful when taking square roots!

If $y = f(x) = ax^2 +bx +c$ is a quadratic function in standard form, then the graph of $f$ has a $y$-intercept at $(0,c)$. Also, if $a > 0$,
then the parabola opens upward, and if $a<0$, the parabola opens downward.

Have students do Problem 3. Explain to students that just because they see something like $h(x)=-(x-4)(x+4)$ they should not automatically start distributing but instead to do what the Problem tells them to do. In this case they are looking for standard form and so it makes sense to distribute, but in other cases it might make sense to leave the equation alone. This is a really difficult concept for students to understand. Don't worry if at first they don't follow, just keep telling the students that mathematical forms have different uses in different contexts and the context is what they need to work on understanding.

===Find $x$-intercepts by factoring and quadratic formula===

Although factoring is done in 100A and is something all students will have seen in high school, students usually need a review of factoring quadratics. Once in factored form, students have the tendency to want to set the quadratic expression equal to zero and then solve the equation. Emphasize to students that this is what we must do to find the $x$-intercepts of a quadratic function, but simply factoring an expression is different than solving an equation.

'''Note:''' Students will come in with a probably-correct-but-mysterious-and-misunderstood method to find the zeros of quadratics, but we want them to learn the $ac$-method (shown below). This standardizes our expectations and gets rid of some of the nonsensical methods like "slide and divide." You can adjust the following example to fit your style, but make sure that the "algorithm" you teach is essentially the same.

*Example:

 
 
 
 
 

Find the $x$-intercepts of $f(x) = x^2 - 3x-4$.

$a = 1$, $c = -4$, so $ac = -4$.
We need factors of $-4$ that add to give $-3$, so we choose $1$ and $-4$. Then,
\[\begin{array}{rl}
x^2 - 3x - 4 & = x^2-4x + x - 4
& = x(x-4) + 1(x-4)
& = (x+1)(x-4).
\end{array}\]
Now, take $f(x) = (x+1)(x-4) = 0$, and we see that either $x+1 = 0$ or $x-4 = 0$, so $x=-1$ or $x=4$. Thus, $f$ has zeros at $x=-1$ and $x=-4$; equivalently, $f$ has $x$-intercepts $(-1,0)$ and $(4,0)$.

Note that we were able to write $f(x)$ in the form $f(x) = (x+1)(x-4)$. In general, this is called factored form.

The [[factored form]], when it exists, of a quadratic function is
\[
f(x) = a(x-r)(x-s),
\]
where $a \neq 0$ is a constant and $r$ and $s$ are the zeros (or $(r,0)$ and $(s,0)$ are the $x$-intercepts) of $f$.

So for our previous example, we have $a = 1$, $r = -1$ and $s=4$. Note that $-1$ and $4$ are the two zeros of $f$.

Have students do Problem 4. Note that part (a) is already in factored form.

===Introduce the quadratic formula===

Have students solve Problem 5(a). Ask them what went wrong when trying to factor the function. Ask if anyone knows of another way to find the zeros of a quadratic function; hopefully someone will remember the quadratic formula.

In general, the zeros of a quadratic function $f(x) = ax^2+bx+c$, if they exist, are given by
\[
x = \dfrac{-b \pm \sqrt{b^2-4ac }}{2a}.
\]

Feel free to do an example if your class seems nauseous at the sight of the formula.

Have students do Problems 5-6.

Have students try Problem 7. Try not to give too big of hints on this. The students now have factored form and standard form. See if they can decide how to put all of these together to find the formula.

5.6: Combining Transformations

2020-06-01T14:45:34Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== * Combine transformations of functions..."

[[5.5: Horizontal Stretches & Compressions | Prior Lesson]] | [[6.1: Introduction to Quadratic Functions | Next Lesson]]
==Objectives==
* Combine transformations of functions
* Practice the material in Chapter 5

==Important Items==
===Definitions:===



==Lesson Guide==

===Warm-Up===
To continue reviewing the material from this section, have students do some sort of review of all transformations used at this point. A table of possible transformations (similar to the one in the Section 5.4 lesson guide) is useful for students to fill in with their groups, but make sure you are able to check these/go over them on the board so they are correct! Test their knowledge by asking them to fill the table out without notes to start and then once they can't fill in any more, they can use their notes.

Have students do Problem 1. If students have any problems you can refer them to Problem 4 from Worksheet 5.5 (this is basically the same problem).

===Combine transformations of functions===
Have students complete Problem 2, which discusses having multiple vertical transformations. This part is usually more intuitive, so make sure you discuss the formulas students came up with for part (c).

Have students complete Problem 3, which discusses having multiple horizontal transformations. This part is much less intuitive. Part (c) only asks that they write two possible functions (thinking about order of operations), so you should discuss which formula goes with which order.

It may be good to do another example of horizontal transformations.
*Example:
Describe how we can transform the graph of $y=m(x)$ to get the graph of $y=m(3x-6)$. First off, we know that the following transformations are involved:

* Horizontal compression
* Horizontal shift

Since there are multiple horizontal transformations, we know that the order matters (c.f., Problem 3), so we will need to figure out what the order is. There are several ways that you can teach this, which are described below (with pros and cons). We will not be testing students on which method they use, but rather just that they can use a correct method. It may be best to choose one method to use for each example, rather than showing students all three methods.

===Method 1: Order of Operations ===
When there are multiple vertical transformations involved, we follow the order of operations.
When there are multiple horizontal transformations involved, we follow the reverse order of operations.

If we follow order of operations, we should multiply by 3 first, then subtract 6. However, since these are horizontal transformations, we need to follow the reverse order of operations. Therefore, the order of transformations is:
* Horizontally shift right by $6$ units
* Horizontally compress by a factor of $3$

Pros: Easy to teach and memorize
Cons: Does not explain why, easy to mix-up

===Method 2: Standard Form of Transformations===
For constants $A$, $B$, $h$ and $k$, the graph of the function \[y=Af(B(x-h))+k\] is obtained by applying the transformations to the graph of $f(x)$ in the following order:
* Horizontal stretch/compression by a factor of $|B|$
* Horizontal shift by $h$ units
* Vertical stretch/compression by a factor of $|A|$
* Vertical shift by $k$ units

If $A<0$, follow the vertical stretch/compression by a reflection across the $x$-axis.
If $B<0$, follow the horizontal stretch/compression by a reflection about the $y$-axis.

To apply this to our problem, we need to rewrite $y=m(3x-6)$ in the standard form of transformations: \[y=m(3(x-2)).\] Now we can see that $A=1$, $B=3$, $h=2$, and $k=0$. Therefore, the order of transformations is:
* Horizontally compress by a factor of $3$
* Horizontally shift right by $2$ units

Pros: Just requires memorization, how the book teaches it
Cons: Does not explain why, easy to mix-up

===Method 3: Bubble Diagram===
If we think about the function $m(3x-6)$ as a composition of several simple functions (refer back to \S5.1: Function Composition), we can represent it using the following function diagram:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/bapznbhvkb?embed
|width=500
|height=500
|border=0
}}

It's tempting to think that this diagram implies that first we would compress, then shift. However, recall that horizontal transformations occur in the opposite way than you might think. To explain why, suppose that $m(a)=b$. To see how this point transforms to become a point on $m(3x-6)$, we do the following:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/gor04smgqr?embed
|width=500
|height=500
|border=0
}}

Therefore, the order of transformations is:
* Horizontally shift right by $6$ units
* Horizontally compress by a factor of $3$

Pros: Visual representation, explains why
Cons: Hard to teach well

NOTE: Since each transformation in Problem 4 involves at most one vertical and at most one horizontal transformation, note that order doesn't matter yet.

Have students work on Problems 4 and 5. It would be beneficial to have students share answers to 5(c) and (d) at least.

Note that the point of Problem 5 is to help students to recognize that order matters if there are multiple vertical or horizontal transformations.

Do one part of Problem 6 to show students how to draw the graph of a transformation of $f(x)$ given the graph for $y=f(x)$. Have students do the rest of Problem 6.

Make sure that students realize that order does matter when applying multiple vertical or horizontal transformations.

==Comments==

*I would only recommend using the bubble diagrams if you have used this throughout other sections (inverse functions, function composition, etc.)

5.5: Horizontal Stretches & Compressions

2020-06-01T14:45:20Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Recognize that horizontal stretches and compres..."

[[5.4: Vertical Stretches & Compressions | Prior Lesson]] | [[5.6: Combining Transformations | Next Lesson]]
==Objectives:==

* Recognize that horizontal stretches and compressions correspond with changes to the inputs
* Horizontally stretch and compress a function that is given either explicitly or graphically

==Important Items==
===Definitions:===
horizontal stretch, horizontal compression

----

==Lesson Guide==

Notes to the instructor: The main focus on this section should be taking a given function and knowing how the stretches/compressions affect the graph of this function.

===Warm-Up===

Perhaps remind students again of the transformations they have seen up to this point. Reiterate the ideas of changing inputs and outputs when we apply transformations.

===Recognize that horizontal stretches and compressions correspond with changes to the inputs===

Have students do Problem 1. Discuss the differences of the graphs.



Observe that the $x$-intercept values change, but the $y$-intercepts stay the same, which makes sense since only the input is being changed.

===Horizontally stretch and compress a function that is given either explicitly or graphically===

Work with students to fill this part out in their course packet (part (d)).
If $f(x)$ is a function and $k>1$ is a constant, then the graph of
* $g(x)=f\left(\frac{1}{k}x\right)$ '''horizontally stretches''' the graph of $f(x)$ by a factor of $k$,
* $g(x)=f(kx)$ '''horizontally compresses''' the graph of $f(x)$ by a factor of $k$.

If $k<-1$, then the graph of $g(x)$ also involves a reflection of the graph of $f(x)$ about the $y$-axis.

Have students do Problems 2-6.

Have students talk about Problem 7 at their tables. Force each table to make a decision and then appoint someone to write their answer on the board. Use this to lead a discussion.

==Comments==
----

5.4: Vertical Stretches & Compressions

2020-06-01T14:45:10Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== * Recognize that vertical stretches a..."

[[5.3: Reflections & Even and Odd Functions | Prior Lesson]] | [[5.5: Horizontal Stretches & Compressions | Next Lesson]]
==Objectives==

* Recognize that vertical stretches and compressions correspond with changes to the outputs
* Vertically stretch and compress a function that is given either explicitly or graphically

==Important Items==
===Definitions:===

vertical stretch, vertical compression

----

==Lesson Guide==

As you'll see below, there are two approaches for going about how to explain this section to your students. You should choose what will work best for your students.

===Warm-Up===

Depending on when this section occurs (in the past it has been after exam 2), it may be helpful to review all function transformation up to this point:

{| class="wikitable"
|+ Caption: example table
|-
! Function
! Transformation
! Point
! Input/output change?
|-
| f(x)
| Base function
| (-1,3)
|N/A
|-
| f(x)+k
| Shift f(x) up by k units
| (-1,3+k)
| Output
|-
| f(x)-k
| Shift f(x) down by k units
| (-1,3-k)
| Output
|-
| f(x+h)
| Shift f(x) left by h units
| (-1-h,3)
| Input
|-
| f(x-h)
| Shift f(x) right by h units
| (-1+h,3)
| Input
|-
| -f(x)
| Reflect f(x) across x-axis
| (-1,-3)
| Output
|-
| f(-x)
| Reflect f(x) across y-axis
| (1,3)
| Input
|}

Have students do Problem 1. It may be worthwhile to discuss this as a review for everyone.



===Recognize that vertical stretches and compressions correspond with changes to the outputs===

Have students work on Problem 2. Discuss the differences students found in parts (b) and (c)



Observe that the $y$-intercept values change, but the $x$-intercepts stay the same, which makes sense since only the output is being changed.

===Vertically stretch and compress a function that is given either explicitly or graphically===

Work with students to fill this part out in their course packet (part (d)).
If $f(x)$ is a function and $k>1$ is a constant, then the graph of
*$g(x)=kf(x)$ [[vertically stretches} the graph of $f(x)$ by a factor of $k$,
* $g(x)=\frac{1}{k}f(x)$ [[vertically compresses} the graph of $f(x)$ by a factor of $k$.
*If $k<-1$, then the graph of $g(x)$ also involves a reflection of the graph of $f(x)$ about the $x$-axis.

Work through the rest of the problems on the worksheet, pausing to discuss as necessary. Make sure that everyone has the correct answer to Problem 5. There are a few different ways to think about it, but one thing to point out is that $x$-intercepts will never change due to a {\em vertical} stretch or compression. As a way to get them thinking ahead, ask what we might do to the graph that would change $x$-intercepts.

If time allows, let students work on the Synthesis Problem. This will likely be very difficult for students, but very cool if they see the trick! It's also similar to a problem on WeBWorK, so it is nice for them to have seen an example like this.

==Comments==

*I'm not entirely fond of how abruptly this lesson starts. I chose to motivate looking at compressions/stretches by talking about how we have added/subtracted constants to the outputs/inputs of a function, so what happens if we instead multiply/divide constants to the outputs/inputs of a function. -Elizabeth
NOTE: This comment was about the old lesson plan, but is also a good way to motivate the section!

5.3: Reflections & Even and Odd Functions

2020-06-01T14:44:57Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ===Objectives=== *Reflect a function across the $x$- and $y$-..."

[[5.2: Vertical & Horizontal Shifts | Prior Lesson]] | [[5.4: Vertical Stretches & Compressions | Next Lesson]]
===Objectives===

*Reflect a function across the $x$- and $y$-axis.
*Reflect and shift a function in the same problem.
*Students will be able to distinguish functions that are even, odd, or neither even or odd.
*Students will be able to use symmetries (even, odd) of a function.

==Important Items==
===Definitions:===
reflected across the $x-$axis, reflected across the $y-$axis, even function, odd function

----

==Lesson Guide==

In this and the following sections, it's tempting to use a table to describe what transformations are doing. A huge downfall of this is that students may think that they always need a table in order to figure out what the graph of a transformation looks like, but aren't always given explicit functions for these tables to be useful. It's more beneficial to continue with the theme of 5.2 and focus on what's happening to points. Continue to write down sentences of the form "The height of the function $g$ at ''some input'' is ''somehow related'' to the height of $f$ at ''some input''" to help students understand why formulas are the way they are.

===Warm-Up===

Have students do Problem 1.

Note: Many students are going to say they don't know what to do. Keep referring them back in their notes. They know what to do but are used to being spoon-fed the material.

Problem 2 is optional depending on how your time is going.

===Reflect a graph across the $x$- and $y$-axis===

Have students do Problem 3, which asks them to draw the reflected graph and create a table of new values. Make note of students approaching this question differently: some students start with drawing the graph and others start by trying to fill in the table. If you ask students to present, have them discuss their thought process. It may be the case that they looked at ordered pairs, which can help you motivate this process.

Help summarize problem 3 in the following way:
*"The height of $g$ at input $x$ is the negative of the height of $f$ at input $x$". You should write this on the board, and have your students to write it down as well. Notice that this sentence is quite literally what it means for one graph to be the reflection of the other across the $y$-axis. You should record the following observation:
**If $g(x)=-f(x)$, then the graph of $g(x)$ is the graph of $f(x)$ '''reflected across the $x$-axis'''.

*"The height of $h$ at input $x$ is the height of $f$ at input $-x$." When they buy that, make the following observation:
**If $g(x)=f(-x)$, then the graph of $g(x)$ is the graph of $f(x)$ '''reflected across the $y$-axis'''.

Have them work on Problems 4-5.



===Reflect and shift a function in the same problem.===

If you wish, do some examples similar to Problem 6.

Have students do Problem 6.

===Recognize even and odd functions===

Example:

* Some functions look exactly the same when we reflect across the $y$-axis.
For example, consider $f(x)=x^2$: \[f(-x)=(-x)^2=(-1)^2(x)^2=(1)x^2=x^2=f(x).\]
Graph and reflect $f(x)$ to illustrate this point.
* Other functions look exactly the same when we reflect across both the $x$- and $y$-axis.
For example, consider $g(x)=x^3$: \[-g(-x)=-(-x)^3=-(-1)^3(x)^3=-(-1)x^3=x^3=g(x).\]
Graph and reflect $g(x)$ to illustrate this point.

Discuss whether or not a graph could look exactly the same when we reflect across the $x$-axis. Mention that we gave a name to all graphs that have these special properties

A function that satisfies the property $f(x)=f(-x)$ then the graph of $f$ is '''symmetric across the $y$-axis''' and $f$ is called an '''even function'''. That is, inputs $x$ and $-x$ have the same outputs.
A function that satisfies the property $f(x)=-f(-x)$ then the graph of $f$ is '''symmetric about the origin''' and $f$ is called an '''odd function'''. That is, the outputs of inputs $x$ and $-x$ are the negations of one-another.

You may want to do an example of how knowing a function is even or odd can give you more information about points.
*Example:
* Suppose that $f(x)$ is even and $f(3)=6$. What is $f(-3)$?
* Since f(x) is even, we know that f(x)=f(-x), so f(-3)=f(--3)=f(3)=6

Make it clear to your students that in general, graphing $f(-x)$ does ''not'' mean you should expect to get the same graph back, but that functions that satisfy this property are called even functions. Some students confuse the two ideas. To this end, it may be valuable to provide non-examples of even and odd functions to help students understand that being even and odd are ''special'' properties that not all functions have, and that one can determine if a function is even or odd by applying specific reflections and seeing the result.

Have students do Problems 7-10.

Students may be confused about how to find values for $f(0)$ in Problems 7 and 8. Once you feel that your students have had enough time to think about it, you can tell them that odd functions must always pass through the point $(0,0)$. To justify this, you could show them the following calculation:

Suppose $f(x)$ is an odd function. We know that $f(x)=-f(-x)$, and so $f(0)=-f(0)$. However, the only number that is the negative of itself is 0, hence $f(0)=0$. If they are still not seeing it, you may solve for it directly. Letting $y=f(0)$, we have that $y=-y$, so $2y=0$, so $y=0$ by dividing by 2. Therefore every odd function must pass through the origin.

You can then contrast this to even functions by graphing a few examples of even functions, some of which pass through the origin and some that may not. The key take away is that for an even function, we have that $f(0)$ could be any number, as the condition for being even only requires that $f(-0)=f(0)$, which is always true.

5.2: Vertical & Horizontal Shifts

2020-06-01T14:44:43Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== Students will be able to * Interpret what a shift doe..."

[[5.1: Function Composition | Prior Lesson]] | [[5.3: Reflections & Even and Odd Functions | Next Lesson]]
==Objectives==
Students will be able to
* Interpret what a shift does to a point on a function
* Vertically shift a function that is given either explicitly or graphically
* Horizontally shift a function that is given either explicitly or graphically
* Combine horizontal and vertical shifts to transform a function
* Interpret a shift in a real world setting

==Important Items==
===Definitions:===
vertical shift, horizontal shift, explicit formula, implicit formula

===Notes:===
Throughout all of Chapter 5, be sure to reinforce the ideas of \S5.5 (on combining function transformations and that the order of function transformations matter).

Use the language of inputs/outputs when talking about transformations; i.e., horizontal shifts result from a change to the [[input]], whereas vertical shifts result from a change to the [[output]]. Do NOT refer to changes as occurring inside/outside the parentheses, as this can lead to confusion; for instance, $x^2+1=(x^2+1)$, but both represent a vertical shift.
----

==Lesson Guide==
===Introduction===
Most students won’t have an issue getting the main idea of how shifts work. Horizontal shifts can be unintuitive to them at first, but they’ll still pick up on how it works quickly. However, the way this section is taught dictates how students will interpret transformations for the rest of this chapter. In later sections, if students don’t understand how specific types of transformations affect the coordinates of points on the graph, they will find these sections quite difficult and lose the physical interpretation for what transformations are doing to the points on a graph. In fact, when some function $g(x)$ is the transformation of $f(x)$, it would be beneficial if you wrote down sentences of the form "The height of the function $g$ at ''some input'' is ''somehow related'' to the height of $f$ at ''some input''". If students can internalize how to discover these types of sentences, it will allow them to uncover what certain formulas should be. These sentences will be written in more detail in the following lesson plans.

One final thing is that students will struggle finding the explicit formula for functions. You'll want to remind them that these transformations are really just compositions. (Some students may actually not know what we mean by explicit and implicit functions -- don't take this for granted, you should define it for them.)

===Warm-Up===
Introduce the idea of function transformations by using ordered pairs.
*Example: Consider the point (1,2) on a coordinate plane (it may be helpful to draw this point on the board).
Ask students what point you would have if you moved this point up, down, left, and right by a certain number of units.
It would be beneficial for you to talk about what coordinate is changing with every movement of the point.

Have students do Problem 1, which walks them through discovering these transformations on a larger scale.

Use these ordered pairs to discuss the relationship between the functions f(x) and g(x):
* How is the function $g(x)$ different from $f(x)$? Tell students to study the inputs and outputs of points, and ask them which has changed, and how has it changed.
* Write down with your students: "The height of $g$ at $x$ is one more than the height of $f$ at $x$". (''Get students in the habit of writing these sentences from now, these sentences come in handy later''.)
* Write down the implicit formula for $g(x)$ from the prior sentence: $g(x)=f(x)+1$. Write down that this is the '''implicit formula''' for $g(x)$ since it refers to another function.
* What has changed about the points on the graph, the inputs, or the outputs? By what value?
* Note that $g(x) = x^2 + 1$. This is the '''explicit formula''' for $g(x)$, since it doesn't refer to any other functions.

Go through a similar discussion of the relationship between f(x) and h(x).

Use Problem 1 to summarize the definitions for vertical and horizontal shifts.

If $f(x)$ is a function and $k$ is a positive constant, then the graph of
* $y=f(x)+k$ is the graph of $y=f(x)$ '''vertically shifted up''' by $k$ units.
* $y=f(x)-k$ is the graph of $y=f(x)$ '''vertically shifted down''' by $k$ units.
When we do a vertical translation, the output value of the points change by $k$ units accordingly.

If $f(x)$ is a function and $h$ is a positive constant, then the graph of
* $y=f(x-h)$ is the graph of $y=f(x)$ '''horizontally shifted right''' by $h$ units.
* $y=f(x+h)$ is the graph of $y=f(x)$ '''horizontally shifted left''' by $h$ units.
When we do a horizontal translation, the input value of the points change by $h$ units accordingly.

Have students work on Problems 2-7. It may be worthwhile to discuss Problems 4 and 6 to reiterate the implicit/explicit formulas. If you want them to get practice with the basic transformations, perhaps have them start with 4-7 and them go back to 2-3.

Note: You may consider breaking up class more by discussing vertical and horizontal shifts separately and having students work on problems in between topics.

Have students work on Problems 9-10 and use these to discuss how transformations affect domain and range. It may be useful to pull up graphs of these functions using something like Desmos.

Giving more examples of domain and range will be helpful as this shows up frequently on homework. For example, you could say something like "$f(x)$ has domain -2<x<=3 and range -4<=f(x)<10. What is the domain and range of f(x+3)-2?"

===Combine horizontal and vertical shifts to transform a function===

Say something like "We may also combine these two types of transformations. Just remember, vertical transformations change the output value of the points, the horizontal transformations change the input values."

*Example:
The graph of $y=f(x-2)+4$ is the graph of $y=f(x)$ shifted to the right by 2 units and up by 4 units. What transformations give the graph of $y=f(x+3) -2$?

It may also be helpful to apply these transformations to a point, and draw arrows representing how its being shifted while also keeping track of these changes of to the side. For example, if the point (1,1) was on f(x) and you wanted to see what happened to that point after shifting right two units and up four units, you could do this visually on the graph and then write down the following next to the graph: $(1,1)\rightarrow(3,1)\rightarrow(3,5)$. This will help emphasize that when you transform functions, you are really changing the inputs and outputs of the points.

Have students do Problems 8, 11-13.

Note: There is very little chance you will get through all of these problems, tell your students that you are going to pick problems that you think they need to work on the most. They can visit office hours or the MRC to work on the rest of the problems.

*Problems 4 \& 6: Students have a tendency to be able to describe this in words but not translate this to a graph. Look for opportunities to connect their words to graphs.
*Problems 5 \& 7: Many students may feel completely lost on these problems. Tell students to pick some key points and describe how those points are transformed.
*Problem 8: Tell students that this problem has shown up on many exams.
*Problem 12: Again, tell students to pick some key points and describe how those points are transformed.
*Problem 13: Make sure students use meaningful and complete sentences.

<!--'''Option 1''': Have students do Problems 1 and 2.

Problems 1 and 2 demonstrate a reason why we should learn about transformations --- they give us a clear way to determine the domain and range of a function if we know that it’s a transformation of another function with a known domain and range. However, these questions are somewhat misleading as to what we’ll be expecting students to do with transformations. A way to alter this to the advantage of this lesson plan is to have the students use their graphing calculators to graph the functions, then ask them to compare and contrast their graphs and the transformed functions of $\sqrt{x}$ with transformations of $\sqrt{x}$, such as $\sqrt{x}+2.$

Once your class seems to be at a good place, bring the class together, and remark that we’ll be spending this chapter trying to understand how altering an equation for a function changes how it looks. Comment something like "today, we’ll be focusing on how to determine if we’ve moved the graph up, left, down, or right."

'''Option 2''': Try to establish a connection between compositions of functions and shifts:
Graph $f(x)=x^2$. Let g(x)=x+1
Ask students to find f(g(x)) and g(f(x))
Go ahead and ask students to graph these functions. If they're unsure how to, you can tell them to plug in some points. Alternatively, you can just graph the two functions yourself.
Ask students to discuss the relationships between f(x) with f(g(x)) and f(x) with g(f(x))

The benefit of option two is that they make they get to realize how compositions are relevant to this chapter. It's valuable for them to make this connection early on, because students tend to struggle to find explicit forms for functions within the context of transformations, and don't realize that it's really just compositions of functions, something they already practiced doing.->

<!--

===Vertically shift a function that is given either explicitly or graphically===
It is crucial that students understand the difference between changing the input and the output in order to understand fully the difference between vertical/horizontal shifts/stretches.

Draw a coordinate plane on the board. Ask them which axis (horizontal or vertical) corresponds to the input and which corresponds to the output. Explain that a change in input results in a horizontal change to the graph and a change in output results in a vertical change. Relate this insight back to where the input and output occur in the coordinate plane.

*Example:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/s6xwv6e4ek?embed
|width=500
|height=500
|border=0
}}

Now on your axis, draw a point, and label its coordinate values. Have your students discuss with their groups where the point will land if you moved the point up by some number of units, and down some number of units. (This should only take groups 10 to 15 seconds, but you want to be sure students see things for themselves. Not only will this help them on the long run, but this chapter involves spatial reasoning, which we don’t develop all that much in this class and is a hard skill for some.)

Ask your class which value changes --- the input or the output. Comment that what you’ve done is a vertical shift (also known as a translation), and then record that a vertical shift changes the output value.

Now tell your class that we can do this to a function, by thinking of its graph as a collection of points. Do an example of a vertical shift of the graph of an explicit function, $f(x)$, by plotting a couple of points on $f(x)$. Emphasize that change was made to the output by noting the difference in the output values of the points plotted (you may find it helpful to label the coordinate values on a point of $f(x)$ and where it ended up on the translated version). A good example function to use is $f(x)=x^2.$ Your examples should demonstrate both upward and downward shifts by both whole and rational units. You'll also want to explain what explicit and implicit formulas mean in your example.

*Example:
Let $f(x) = x^2$. Note that (1,1) and (-2,4) are points on this graph, let’s plot these.
* Tell your students we're going to translate the points, and the whole graph, up two units. Draw the new graph, and label the translated points. You may want to draw an arrow showing how the points traveled.
* How is the function $j(x)$ different from $f(x)$? Tell students to study the inputs and outputs of points, and ask them which has changed, and how has it changed.
* Write down with your students: "The height of $j$ at $x$ is two more than the height of $f$ at $x$". (''Get students in the habit of writing these sentences from now, these sentences come in handy later''.)
* Write down the implicit formula for $j(x)$ from the prior sentence: $j(x)=f(x)+2$. Write down that this is the '''implicit formula''' for $j(x)$ since it refers to another function.
* What has changed about the points on the graph, the inputs, or the outputs? By what value?
* Note that $j(x) = x^2 + 2$. This is the '''explicit formula''' for $j(x)$, since it doesn't refer to any other functions.
* If $l(x) = f(x)-1.5$, what changes about the graph? What changes about the points?
{{#widget:Iframe
|url=https://www.desmos.com/calculator/1tzyms9bsp?embed
|width=500
|height=500
|border=0
}}

It’s tempting to make a table and plot the values to help convince students, but all this conveys is that if you want to figure out a transformation, you need to make a table. They tend to lose sight of the big picture about how adding some constant to a function is itself telling you what happens to the graph. This is why focusing on which coordinate of the point changes is more productive. On the other hand, the sentence "The height of $j(x)$ at x is the two more then the height of f at $x$" tells you exactly what the transformation is.

Have students do Problems 3, 5, and 6. It may be better to have students do 5, then 6, then 3, so that they practice the concepts of translations first before applying it to a real-world problem. Make sure you've defined explicit formula before having them do problem 5.

===Horizontally shift a function that is given either explicitly or graphically===

Do an example of a horizontal shift of the graph of an explicit function, $f(x)$. Emphasize that change was made to the input. A good example function to use is $f(x)=x^2.$ Your examples should demonstrate both left and right shifts by both whole and rational units. Note that students will see everything as backwards here. But be sure it’s justified why things happen the way they do.

*Example:
Let $f(x) = x^2$.
* Lets plot (1,1) and (-2,4) again.
* Tell your class you're now going to translate the graph, and the points, left two units.
* Graph the new function and label the translated points. Drawing arrows between the points may be helpful to show students what you've done.
* How is the function $g(x)$ different from $f(x)$? Tell students to study the inputs and outputs of points, and ask them which has changed, and how has it changed.
* Write down with your students: "The height of $g$ at $x$ is the height of $f$ at $x+2$." (''You may need to use some input values to convince them of this, they may think it's backwards''.)
* From the prior statement, derive the implicit equation for $g(x)$: $g(x)=f(x+2)$. (''Maybe remark here that this is an '''implicit formula''' for'' $g(x)$, ''just to remind them of this word.'')
* Note that $g(x) = (x+2)^2$. (''Maybe remark here that this is an '''explicit formula''' for'' $g(x)$, ''just to remind them of this word.'')
* If $h(x) = f(x-1.5)$, what changes about the graph? What changes about the points?

{{#widget:Iframe
|url=https://www.desmos.com/calculator/myefcpk6ir?embed
|width=500
|height=500
|border=0
}}

Students will likely have difficulty telling which form $g(x-h)$ or $g(x+h)$ corresponds to a left/right shift. One method is to plug in $h$ into g(h-h)=g(0), to demonstrate that the vertex of the parabola in the above example must now have x value h, instead of 0. Then do a similar thing with $-h$ and $g(x+h)$. Alternatively, have them check it by graphing a few functions to test their intuition and then have them try to explain why that might be. It will be tricky for them, but given practice, they will get it. However, the hope is that the sentences like "The height of $g$ at $x$ is the height of $f$ at $x+2$" will help them be convinced of which formula should be left shifts, and which should be right.

If $f(x)$ is a function and $h$ is a positive constant, then the graph of
* $y=f(x-h)$ is the graph of $y=f(x)$ '''horizontally shifted right''' by $h$ units.
* $y=f(x+h)$ is the graph of $y=f(x)$ '''horizontally shifted left''' by $h$ units.
When we do a horizontal translation, the input value of the points change by $h$ units accordingly.

Have students do Problems 4, 7, and 8. As before, you may find it more beneficial to have them do 7, then 8, then 4. ->

5.1: Function Composition

2020-06-01T14:44:29Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Identify the inputs and outputs for a function th..."

[[4.4: Applications of the Logarithm | Prior Lesson]] | [[5.2: Vertical & Horizontal Shifts | Next Lesson]]
==Objectives:==
* Identify the inputs and outputs for a function that is the composition of two functions
* Evaluate compositions of functions at given inputs
* Find the formula for a composite function $f(g(x))$, given formulas for $f(x)$ and $g(x)$
* Find two functions $f$ and $g$ such that $h(x)=g(f(x))$, given $h(x)$

===Notes:===
We will NOT cover the domains of compositions of functions. Composition of functions is not easy for students. We worked with composition in 2.2, so we continue this and will also introduce '''decomposing''' functions. (As we all know, the ability to recognize functions as compositions is immensely important when it comes time to work with the chain rule in Calculus.) Students might have some familiarity with solving for actual formulas, but algebra mistakes and careless use of parentheses can cause real problems. Furthermore, they generally have not dealt with composite functions in tables and graphs, and they often struggle to have a deeper conceptual understanding of this section.

----

==Lesson Guide==

Start class by talking about the goal of this section: We want to transform graphs by translating, reflecting, compressing, and stretching. Students may not be aware about what you mean by these transformations. A quick 2 minute demo with Desmos could help with this. (Try [http://www.desmos.com/calculator/owwksceg0l this] premade demo.) Remark that these transformations can be understood through the perspective of compositions of functions, and so you'll spend 5.1 getting comfortable with compositions, and then you'll spend the rest of chapter 5 focusing on how to actually make these transformations happen for any given function. The demo is also included below:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/s1zwlq48w9?embed
|width=500
|height=500
|border=0
}}

===Warm-Up===

'''Option 1''': Have students do Problem 1

'''Option 2''': Have students remember what they did in \S2.2:
Let $f(x)=x+1$ and $g(x)=x^2+5$. Find
1. f(3)
2. g(4)
3. g(f(x))
4. g(f(3))

One major issue that some instructors have found with problem 1 is that it's disconnected from the rest of the section. As mentioned in the objectives, this section focuses on them finding compositions, trying to evaluate compositions, and then decomposing functions. Although we want them to be able to understand how to compose two functions given a written description of them, it's not going to be very natural for them to figure this out as the first problem of the day. You might consider saving this problem for the end, once they have a deeper understanding of the relationship between inputs and outputs with compositions. To this end, a more natural way to start this section is to just remind them that they've already seen compositions before by having them do option 2, described above. Note that if you do option 2 that you may consider skipping the next part of this lesson plan.

===Identify the inputs and outputs for a function that is the composition of two functions===

Initially, it may be beneficial to remind students of \S2.2. Revisit that worksheet and do some examples similar to those on Worksheet 2.2. It is probably necessary to show students the arrow diagram found in Lesson Plan 2.2 again. Then, do several examples similar to Problem 1 on Worksheet 5.1 while utilizing the arrow diagram. Your examples should emphasize giving ``practical interpretations'' of the functions as a complete sentence.

* Example:
 
 
 
 
 

* Example:
 
 
 
 
 

Have students do Problem 2.

===Evaluate compositions of functions at given inputs===

While the concept of Problem 3 is very easy, students struggle with the table having more than one function's outputs listed. Show them that the table is just a consolidation of what would be three separate tables for the functions $p(x),$ $q(x),$ and $r(x).$ Do part of Problem 3 to show them how to use the consolidated table. You may also want to do part of Problem 6 to remind them how to use graphs in a function composition.

Have students do Problem 3, 5, and 6.

Have groups of students present their answers to Problems 3 and 6.

===Find the formula for a composite function $f(g(x))$, given formulas for $f(x)$ and $g(x)$===

Have students do Problem 4.

Remind students to use parentheses when plugging the expression for $f(x)$ into $g(x)$ to obtain the correct formula for a function composition.

===Find two functions $f$ and $g$ such that $h(x)=g(f(x))$, given $h(x)$===

Now tell the students that we are going to start working backwards. Begin with a simple example of a function that can be decomposed as the composition of two nontrivial functions in more than one way. On the board, do one of the ways, and then ask students if they can find another possible decomposition.

* Example:
 
 
 
 
 

Have students do Problems 7 and 8. Explain to students that there are multiple correct answers to Problem 8.

Since there are multiple correct answers to Problem 8, use the remaining class time to have groups put solutions on the board.

===Comments===

My students found making a bubble diagram outlining the order of operations and then making a chart of possible decompositions of the function useful. For example, let $f(g(x))=h(x)=\sqrt{5x^3-3}$. The operations, in order, are cube, multiply by 5, subtract 3, and take a square root. $g$ can include any number of these operations starting from the inside and working out.

$\begin{array}{c|c|c} f(x) & g(x) & f(g(x)) \\
\sqrt{5x^3-3} & x & \sqrt{5x^3-3} \\
\sqrt{5x-3} & x^3 & \sqrt{5x^3-3} \\
\sqrt{x-3} & 5x^3 & \sqrt{5x^3-3} \\
\sqrt{x} & 5x^3-3 & \sqrt{5x^3-3} \\
x & \sqrt{5x^3-3} & \sqrt{5x^3-3} \\
\sqrt{x+2} & 5x^3-5 & \sqrt{5x^3-3} \\ \end{array} $

The last line of the table is useful for some of the WeBWorK problems and to let the students know that there are infinitely many possibilities. I would label the first and fifth decompositions with cheese so that the students know not to use any trivial decomposition in their solutions. (Morgen Bills)

4.4: Applications of the Logarithm

2020-06-01T14:44:15Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== * Identify the domain and range of $t=a\log(P-b)$ and $t..."

[[4.3: Logarithms & Exponential Models | Prior Lesson]] | [[5.1: Function Composition | Next Lesson]]

==Objectives==

* Identify the domain and range of $t=a\log(P-b)$ and $t=c\ln(P-d)$ and understand their relationship to the domain and range of $P=10^{t/a}+b$ and $P=e^{t/c}+d$.
* Manipulate real-world examples that can be modeled by a logarithm

----

==Lesson Guide==
[[Kelsey's Alternative Approach]]

Whether you use Kelsey's approach or this approach, consider at some point using the following demo to show how the log base changes the graph (this app also exists in the book in 8.2, which may actually be better to use):

{{#widget:Iframe
|url=https://www.desmos.com/calculator/qqwbqeqnos?embed
|width=500
|height=500
|border=0
}}

===Warm-Up===

Have students do Problems 1 and 2.

===Identify the domain and range of $t=a\log(P-b)$ and $t=c\ln(P-d)$ and understand their relationship to the domain and range of $P=10^{t/a}+b$ and $P=e^{t/c}+d$.===

Graph several logarithmic functions of the form $t=a\log(P-b)$ and $t=c\ln(P-d).$ Ask students to comment on the common characteristics of these graphs. Ask them to find the exponential function for which they are the inverse. Compare the graphs of the exponential with the graph of its inverse to help students gain intuition on their relationship.

-Example:
 
 
 
 

-Example:
 
 
 
 

Emphasize that the domain and range of a function are switched for that function's inverse. Show this carefully for an exponential and its inverse function.

Discuss asymptotes, and remind students of the fact that an exponential of the form $P=(b)^t$ has a horizontal asymptote at $P=0.$ So a logarithmic function of the form $t=\log_b(P)$ has a vertical asymptote at $P=0.$ Generalize this to asymptotes of other logarithmic functions like $a\log(P-b)$.

*Note: There are no problems in this section asking students about graphs or asymptotes. Consider asking students to graph one or more of the functions they find in these problems, and ask them to interpret the meaning of the asymptote in the context of the problem.

Have students do Problem 3.

===Manipulate real-world examples that can be modeled by a logarithm===

While we don't formally introduce the idea of orders of magnitude, students still struggle with the idea of how to compare two quantities. One way you can help students understand this is by doing an example that just focussing on calculating how many times more one quantity is than another.

-Example:
Suppose that Susan has saved \$10,000 and Marc has saved \$6,300. How many times more money has Susan saved than Marc?
\begin{align*}
6300x&=10000 \\
x&=\frac{10000}{6300} \\
x&\approx1.5873
\end{align*}
What if all we knew was that Susan has saved $D_S$ dollars and Marc has saved $D_M$ dollars? How would we compute how many times more money Susan saved than Marc?
\begin{align*}
D_Mx&=D_S \\
x&=\frac{D_S}{D_M}
\end{align*}

Have students do Problems 4 and 5. Students may not realize that they will need the formula given to them in Problem 4 in order to do Problem 5.

Since students often struggle with using $A_0$ in Problem 6, let students know that $A_0$ is \textbf{not} a variable, but an unknown, constant quantity.

Have students do Problem 6.

4.3: Logarithms & Exponential Models

2020-06-01T14:44:00Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Given an exponential model and an output, use logari..."

[[4.2: Properties of Logarithms | Prior Lesson]] | [[4.4: Applications of the Logarithm | Next Lesson]]
==Objectives:==

* Given an exponential model and an output, use logarithms to solve for input.
* Compute doubling time and half-life
* Convert between the forms $f(t)=a(b)^t$ and $f(t)=ae^{kt}$

----

==Lesson Guide==

===Warm-Up===

Have students do Problem 1.

Guide students through problem 2

===Given an exponential model and an output, use logarithms to solve for input.===

Introduce an exponential model in the context of a word problem.

-Example:
Use an exponential model where the inputs are years and the outputs are sizes of a population.
 
 
 
 

Ask students to help determine how given a desired population, one can find the year in which that population is achieved. Connect this idea back with the notion of logarithms as inverse functions of exponentials. Emphasize the importance of writing the final answer in a complete sentence with correct units.
 
 
 
 

While solving, do every step algebraically and THEN put the final answer into a calculator. Tell students that if they enter each step into their calculator and round that answer, the final answer may be way off. Do not round until the final step, if at all.

Have students do Problems 3 and 4.

===Compute doubling time and half-life===

Do some examples given as word problems that include the phrases "doubling time" and "half-life"

-Example:
 
 
 
 

-Example:
 
 
 
 

Have students do part (a) of Problem 6

===Convert between the forms $f(t)=a(b)^t$ and $f(t)=ae^{kt}$.===

Explain to students that given a growth factor $b$, we can always rewrite $b$ as $e^k$ for some $k$ using logarithms. In doing so, we can convert an exponential model the form $f(t)=a(b)^t$, which clearly exhibits the annual growth rate, to an equivalent model $f(t)=ae^{kt},$ which clearly exhibits the continuous growth rate. Do an example of this.

-Example:
 
 
 
 

Conversely, given a continuous growth rate $k$, we can always find a growth factor $b$ such that $b=e^k$ by simply evaluating $e^k.$ Then, we can determine what the effective annual growth rate, $b-1$, is. In doing so, we can convert an exponential model $f(t)=ae^{kt}$ to an equivalent model $f(t)=a(b)^t.$ Do an example of this.

-Example:
 
 
 
 

Students often struggle with this concept. Emphasize to students that the two exponential expressions model identical populations, but the format $f(t)=ae^{kt}$ clearly shows the continuous growth rate while $f(t)=a(b)^t$ clearly shows the effective annual growth rate.

Have students do Problems 7-8.

Throughout class, pause and have students present their solutions. Try to ensure that all students feel comfortable with the basic concept of the relationship between logarithms and exponential models.

Do the Focus Problem.

==Comments==
----
I found it hard to motivate why we would want to convert between between annual and continuous models. I gathered the following reasons from a conversation with Nathan. Feel free to modify or add to these.
* In finance, biology, or chemistry, different models may be used in different scenarios. The ability to convert between the two models could help them better compare models that arise if they appear in different forms.
*It's valuable for students to be able to tell the differences between continuous versus annual models so that they can understand how and why the growth factors/rates are different. For example, since continuous growth models grow faster, the continuous rate should always be expected to be smaller than the annual rate.

4.2: Properties of Logarithms Part II

2020-06-01T14:43:49Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Understand and practice using common log a..."

[[4.2: Properties of Logarithms Part I | Prior Lesson]] | [[4.3: Logarithms & Exponential Models | Next Lesson]]
==Objectives:==

* Understand and practice using common log and natural log
* Discover and utilize properties of logarithms
*Use properties of logarithms to solve exponential equations.
*Use Properties of exponentials to solve logarithmic equations.

==Important Items==
This lesson is a continuation of [[4.2: Properties of Logarithms Part I]]
----

Have students work on problem 1 as a sort of warm-up. You may need to remind them of some of the properties as they work.

Do some examples similar to Problems 2 '''In your examples, it is ''imperative'' that you use language that enforces that logs are not something like a number you multiply by, it's a ''function'' you ''compose'' with both sides. Saying something as literal as "And now we compose both sides with log by plugging both sides into log" would be productive. Otherwise, students may do things like distribute the word 'log' or divide the word alone to the other side.'''

-Example:
 
 
 
 

-Example:

 
 
 
 

Have students do Problem 2.

Have students work on problem 3. Note: after asking them to work on it for 3 minutes you should bring everyone together and briefly lead a discussion to get 6e^{0.012t}=8 written on the board.

Have students work on problem 4. Be prepared to remind a few tables of the proper equations to use.

==Comments==
----
* Note that for problem 2e, $x>0$. Be aware that if students naively apply the properties they might get to $x^4 = 10^4$ and conclude $x = \pm 100$.
* Students tend to struggle to understand logarithms as a function. Be very careful with the language you use while solving for an input using a log. Saying something like "apply log to both sides" might come off as "multiply by log on both sides" and this will cause students to lose all understanding of log as a function. To enforce the function aspect, I suggest saying "compose both sides with the logarithm function." In examples, it may be better to avoid what the workbook does, and do something like "Evaluate log_3(x) at x=9", to emphasize that log is really a function.
* Instead of doing problems similar to problem 1 in 8.2, it may better to do an example: "What is 10^log(100)?" Walking them through this may more naturally lead them to see that log(100) is the power of 10 that gives 100.

4.2: Properties of Logarithms

2020-06-01T14:43:36Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Understand and practice using common log..."

[[Interlude: Introduction to Logarithms | Prior Lesson]] | [[4.2: Properties of Logarithms Part II | Next Lesson]]
==Objectives:==

* Understand and practice using common log and natural log
* Rewrite and evaluate statements with logs using exponents
* Rewrite and evaluate statements with exponents using logs.
* Discover and utilize properties of logarithms

==Important Items==
===Definitions:===

common log, natural log
----

===Understand basic properties of logarithms and how to evaluate them===

Introduce logs as the [[inverse functions]] of exponentials. Then use an example like the following to talk about how to find the inverse of exponential functions.

- Example:
Suppose $P=Q(t)=100(2)^t$.
* If $Q(0)=100$, then what is $Q^{-1}(100)$?
* Solve $Q^{-1}(P)=3$ by using the given equation for $Q(t)$.

We can continue to use $Q(t)$ to evaluate and solve for $Q^{-1}(P)$, but would be nice if we could write an equation for $Q^{-1}(P)$. In fact, we can do that as long as we can find a way to "undo" $Q(t)$ by working backwards.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/rditdvdrxe?embed
|width=750
|height=750
|border=0
}}

Students tend to think that roots are the inverse of exponentials. Make sure to address and emphasize the difference between the two.

The purpose of today's lesson is to learn how to "undo" this final step.

[[logarithmic function]] is the inverse function of the special exponential
function, $Q(t)=b^t$. In other words, if $P=b^t$, then $\log_b(P)=t$.

Do several examples of how to evaluate logs. Also graph several of these examples so that students gain an understanding of the graphic qualities of a logarithmic function and how it relates to its inverse exponential function.

* Example:
 
 
 
 

* Example:
 
 
 
 

Have students do Problems 1-3.

Throughout class, pause and have students present their solutions. Try to ensure that all students feel comfortable with the basic concept of a logarithm being an inverse of an exponential.

===Understand and practice using common log and natural log===

Explain that base 10 and base $e$ are so common that we developed specific mathematical notation to denote when the base is either 10 or $e$.

* $\log(P)$ is the power of 10 that gives $P$. So if $t=\log(P)$, then $10^t=P$.
* $\ln(P)$ is the power of $e$ that gives $P$. So if $t=\ln(P)$, then $e^t=P$.

Some students forget about the base matters, since we "drop" it when we use the common and natural log notations. Emphasize that other bases can be used, but $\log(P)=\log_{10}(P)$ and $\ln(P)=\log_e(P)$.

Do an example similar to Problem 4.

-Example:
 
 
 
 

*Have students do Problems 4 and 5.

==Rewrite and evaluate statements with logs using exponents.==

*Do an example similar to Problem 6.

-Example:
 
 
 
 

*Have students do Problem 6.

===Rewrite and evaluate statements with exponents using logs.===

*Do some examples similar to Problems 7 and 8.

-Example:
 
 
 
 

Example:

 
 
 
 

*Have students do Problems 7 and 8

===Discover and utilize properties of logarithms.===

Have students do Problem 9.

Use Problem 9 to talk about the following properties of logarithms.

Properties of Logarithms:
* $\log_b(xy)=\log_b(x)+\log_b(y)$
* $\log_b\left(\frac{x}{y}\right)=\log_b(x)-\log_b(y)$
* $\log_b(x^y)=y\log_b(x)$
* $\log_b(b^x)=x$
* $b^{\log_b(x)}=x$

To explain why the first three properties are true, relate them to the corresponding properties of exponentials. To explain why the last two properties are true, talk about the connection between inverse functions (it may be helpful to draw bubble diagrams).

Interlude: Introduction to Logarithms

2020-06-01T14:43:24Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== * Understand basic properties of logarithms and how to evaluate th..."

[[4.1: Inverse Functions | Prior Lesson]] | [[4.2: Properties of Logarithms | Next Lesson]]

==Objectives==

* Understand basic properties of logarithms and how to evaluate them

==Important Items==
===Definitions:===
logarithmic function

===Notes:===

This lesson is considered to be the necessary background to \S4.2. Logarithms are difficult for many students! Furthermore, while the remainder of Chapter 4 focuses on '''only''' common logs and natural logs, we will discuss logs with any base in this lesson.

----

===Understand basic properties of logarithms and how to evaluate them===

Introduce logs as the [[inverse functions]] of exponentials. Then use an example like the following to talk about how to find the inverse of exponential functions.

- Example:
Suppose $P=Q(t)=100(2)^t$.
* If $Q(0)=100$, then what is $Q^{-1}(100)$?
* Solve $Q^{-1}(P)=3$ by using the given equation for $Q(t)$.

We can continue to use $Q(t)$ to evaluate and solve for $Q^{-1}(P)$, but would be nice if we could write an equation for $Q^{-1}(P)$. In fact, we can do that as long as we can find a way to "undo" $Q(t)$ by working backwards.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/rditdvdrxe?embed
|width=750
|height=750
|border=0
}}

Students tend to think that roots are the inverse of exponentials. Make sure to address and emphasize the difference between the two.

The purpose of today's lesson is to learn how to "undo" this final step.

[[logarithmic function]] is the inverse function of the special exponential
function, $Q(t)=b^t$. In other words, if $P=b^t$, then $\log_b(P)=t$.

Do several examples of how to evaluate logs. Also graph several of these examples so that students gain an understanding of the graphic qualities of a logarithmic function and how it relates to its inverse exponential function.

* Example:
 
 
 
 

* Example:
 
 
 
 

Have students do Problems 1-3.

Throughout class, pause and have students present their solutions. Try to ensure that all students feel comfortable with the basic concept of a logarithm being an inverse of an exponential.

==Comments==
----
*Make sure students are aware that from 4.2 on, they will be expected to only use common log and natural log. The purpose of the interlude was to introduce logs in a general sense so that they don't think $10^x$ and $e^x$ are the only exponential functions with inverses, but logs with different bases won't be used for the rest of the semester.

*In my class, although inverses had come up regularly in the past few weeks, students are still not completely comfortable with them. Writing down in a box to the side the fact that $Q(t) = P$ and $Q^{-1}(P) = t$, and showing how this was used in defining the logarithm via color coding really seemed to get the point across that the logarithm is the inverse of the exponential and also helped highlight the relationship between the two (specifically using the exponential function to solve logarithmic expressions). -Elizabeth C

4.1: Inverse Functions

2020-06-01T14:43:09Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Use the horizontal line test to determine if a function i..."

[[3.5: Continuous Growth | Prior Lesson]] | [[Interlude: Introduction to Logarithms | Next Lesson]]
==Objectives:==
* Use the horizontal line test to determine if a function is invertible
* Find the explicit formula for the inverse and give interpretations of the inverse function

==Important Items==
===Definitions:===
invertible, inverse function
===Notes:===
In this lesson, we will NOT be teaching students to sketch an inverse function by reflecting over the line <math>y=x</math> or to verify that a function is an inverse function by using the fact that <math>f(f^{-1}(x))=x</math>. This notion does come up on problem 7 but you can have them discover this on their own.

Also, in high school a lot of students will have learned a procedure
similar to the following for finding a formula for the inverse of a function:

* Start with the formula of the function,
* Swap the <math>x</math>'s and the <math>y</math>'s in this formula.,
* Manipulate to make <math>y</math> the subject, i.e. "solve for <math>y</math> in the new formula".

The advantage of swapping is that students are more accustomed to solving for <math>y</math> and so are less likely to become confused as to what algebraic manipulations are needed in Step 3. The (huge) disadvantage is that if the <math>x</math> and <math>y</math> actually stand for real-world quantities then you also have to swap the meaning of <math>x</math> and <math>y</math> when you calculate the formula for the inverse. This is a tough point for a lot of students. Namely, if <math>x</math> is in dollars and <math>y</math> is in apples in the original function, then they expect <math>x</math> to be in dollars and <math>y</math> to be in apples in the inverse as well. It is for this reason that we advocate an approach to computing inverses in which you simply re-arrange an equation.
Hence you must be careful to imitate follow the examples given here.
----

==Lesson Guide==

===Warm-Up===

Have students do Problems 1 and 2.

===Use the horizontal line test to determine if a function is invertible===

We say a function is [[invertible]] if its inverse is also a function.

Remind students the definition of a function.
A function is a relation (or a rule) that assigns each input to only one output.

{{Ex-begin}}
Walk students through problem 3 as an example. This is a very helpful way to teach students how to find the explicit formula of the inverse function because it emphasizes the order. Using the bubble diagram is also useful in other areas of the course.

Let <math>f(x) = 3\sqrt{x+1}-4</math>. When we are computing what happens to <math>x</math>, we first add 1 to <math>x</math>, then take the square root, and so on.

a) Represent this chain of computations visually by labeling the arrows in the diagram below.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/x9qlb082jn?embed
|width=500
|height=500
|border=0
}}

https://www.desmos.com/calculator/x9qlb082jn

b-c) Represent this chain of computations visually by labeling the arrows in the diagram below.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/jcprbs1x4m?embed
|width=500
|height=500
|border=0
}}

https://www.desmos.com/calculator/jcprbs1x4m

d) Allow students to try to find the inverse of the invertible functions in Problem 3. The next objective will delve into this more deeply, and students may need to go back to these problems after seeing Objective (ii) in order to finish them.

At the end, introduce the horizontal line test as a way to verify whether or not a graph represents a function. Talk about how this is connected to the vertical line test and the graph of a function. It may be helpful to talk about the graph of the function <math>f(x) = 3\sqrt{x+1}-4</math>.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/fuxw04ouyy?embed
|width=500
|height=500
|border=0
}}

https://www.desmos.com/calculator/fuxw04ouyy

{{Ex-end}}

===Find the explicit formula for the inverse and give interpretations of the inverse function===

Have students do Problem 4.

Given a function <math>P=f(Q),</math> the notation for the inverse function will be <math>Q=f^{-1}(P).</math> Emphasize to your students that the inverse notation does not mean ``reciprocal." Furthermore, make sure your students understand that the inputs and outputs, or the domain and range, of an invertible function become the outputs and inputs, or the range and domain, respectively, of the inverse function.

Have students do Problems 5-8.

==Comments==
----
* I found it productive to use the second warm-up problem to transition into what an invertible function. We talked about it as a class first, put up an example, and then I defined invertible functions.
* It may be good to have an example of an example that interprets the meaning of an inverse for an exponential function, as a way to motive 4.2: Logarithms. You could end the example by saying something like: "And now next time, we'll figure out how to find the inverse of an exponential explicitly."
* Note: Problems 6 and 8 present functions that are only invertible on restricted domains (Problem 8 gives the restricted domain). I think it would be helpful to scaffold this idea a little bit more than what is present in the lesson plan. For example, with Problem 6 part (b) I think it'd be helpful to have a whole class discussion about why it makes sense not to include the negative root (i.e., we don't want to have negative values for "t" because we only know what the population is after 1982, not before), and so in fact this function can be made invertible if we restrict our domain to nonnegative values for time.
*A HEAD'S UP: I wish I had paid more attention to the bubble diagrams/emphasized them a little bit more when I taught this course. The diagrams pop up again in 5.1 (Compositions), and can be useful for explaining why the order of transformations makes sense (5.6). Although, if you haven't been stressing these sorts of diagrams by the time your class gets to 5.6, it can feel a little abrupt to start heavily emphasizing them in that section. TLDR: plan accordingly, make sure to look ahead to the lesson plan for 5.6.

3.5: Continuous Growth

2020-06-01T14:42:07Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Introduce the notion of compounding continuously and the number <math>e</..."

[[3.4: Compound Growth | Prior Lesson]] | [[4.1: Inverse Functions | Next Lesson]]
==Objectives:==

* Introduce the notion of compounding continuously and the number <math>e</math>
* Understand the difference between annual and continuous growth rates

==Important Items==
===Definitions:===

continuous growth

==Lesson Guide==

===Warm-Up===

Have students do Problem 1

===Introduce the notion of compounding continuously and the number <math>e</math>===

'''Introduction''': In the last lesson, we learned about compound interest and looked at how investments can grow with interest compounded annually, monthly, weekly, or even daily. We noted (hopefully!) that as interest of a given rate is compounded more and more frequently, we earn more money overall, since the interest earned in one period earns interest itself in the next. This raises an interesting question: How much money could be earned if we compounded interest non-stop? Let's build a chart and see what happens as we make the compounding period smaller and smaller.

{{Ex-begin}}
Choose a principal, <math>P=1</math>, and an interest rate, <math>r=1</math>, for an account. Have the students help to compute the value of the account after one year if the interest is compounded <math>n</math> times per year. Recall from the previous lesson that the amount one year later (i.e., when <math>t=1</math>), will be <math>A(1)=P\left(1+\frac{r}{n}\right)^{n}.</math>

'''Note:''' if you choose <math>P=1</math> and <math>r=1,</math> increasing <math>n</math> will make the value of the account after one year converge to <math>e.</math>

{| class="wikitable"
|-
|Compound frequency
|Value of account after one year
|-
|n=1
|2
|-
|n=2
|2.25
|-
|n=4
|2.441406
|-
|n=12
|2.613035
|-
|n=365
|2.714567
|-
|n=8760 (hourly)
|2.718127
|-
|n=525,600 (each minute)
|2.718279
|-
|n=31,536,000 (each second)
|2.718282
|}

Discuss how compounding at a higher and higher frequency starts to appear as though you are "always" compounding, i.e., compounding continuously.

What does this table show us? You may find it interesting to note that while the number of times we compounded got bigger and bigger, the amount we earn gets bigger too - but not too big! We never ended up with more than 2.75 dollars in our account. Explain to your students that this value, 2.718282, is very close to the number <math>e</math> and that we use <math>e</math> to represent continuous growth, or in this case, interest compounded continuously.

{{Ex-end}}

If a bank has an annual interest rate of <math>r</math> that is [[compounded continuously]],
the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit of the
principal <math>P</math> is <math>A(t)=Pe^{rt}</math>

Have students do Problems 2-4.

===Understand the difference between annual and continuous growth rates===

For many instructors these are terms you may have not heard before. Make sure to look up these terms in the book so that you know what they are!

Unless the problem says [[continuous growth rate]], the student should assume it is not continuous.

Do at least one example that explores continuous vs. non-continuous growth rates.

{{Ex-begin}}
As of 2011, Lincoln, NE, has a population of 262,341 people. Suppose the population grows at a continuous growth rate of 2.1% per year.

*Find a formula for <math>P(t)</math>, the population of Lincoln, NE, <math>t</math> years after 2011.
'''Answer:''' <math>P(t)=262341e^{0.021t}</math>.
*By what percent does the population increase each year?
'''Answer:'''
<math>
262341e^{0.021t}=262341(1+r)^t
</math>

<math>
262341e^{0.021}=262341(1+r), t=1
</math>

<math>
e^{0.021}=1+r
</math>

<math>
e^{0.021}-1=r
</math>

<math>
0.02122\approx r
</math>

So the city would be growing at approximately 2.122% each year.
*Under this model, predict the population in 2020.
'''Answer:''' <math>P(9)=262341e^{(.021)(9)}=316918.6715\approx 316,918</math>

{{Ex-end}}

Be sure to make a clear distinction between the [[continuous growth rate]] of 2.1\% and the [[growth rate]] of 2.122\%. (Remember nominal vs. effective interest rates? It's sort of the same idea.)

Have students do Problems 5-6.

Tie the sections of Chapter 3 together by comparing the formulas in Sections 3.4 and 3.5 with an exponential function of the form <math>f(t)=a(b)^t.</math> Have students help you fill in the following table:

{| class="wikitable"
|-
|Exponential Formula
|Initial Value
|Growth Factor
|-
|<math>f(t)=a(b)^t</math>
|
|
|-
|<math>A(t)=P{\underbrace{(1+r)}_{b}} ^t</math>
|
|
|-
|<math>A(t)=P{\underbrace{\left[\left(1+\frac{r}{n}\right)^n\right]}_{b}} ^{t}</math>
|
|
|-
|<math>A(t)=P{\underbrace{(e^r)}_{b}} ^t</math>
|
|
|}

Remind students how the growth factor relates to the effective annual rate (or annual growth rate). Do an example where you find a formula <math>A(t)=P(b)^t</math> given continuous growth rate <math>A(t)=Pe^{rt}</math> (i.e., solve for <math>b</math> given <math>e^r</math>) and hint that we may want to do the reverse as well, which we'll talk about in Chapter 4 (i.e., solve for <math>r</math> given <math>b</math>).

*Work Problem 7 in groups. Let them move on to the Synthesis Problem only if time allows. (But it is a good review problem and a nice teaser for anyone going on to calculus!)

Work on the Focus Problem. ''This will likely be challenging for students as they have not seen functions like this before. After they have had some time to get comfortable with the function and its graph, lead them to applying terms such as "horizontal asymptote" to what they are seeing. In particular, note that this model might be more realistic than others we have considered because populations tend to reach the "carrying capacity" of their environment and stabilize at that population.

==Comments==
----

3.4: Compound Growth

2020-06-01T14:41:49Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Introduce (annual) compound interest * Compound interest m..."

[[3.3: Graphs of Exponential Functions | Prior Lesson]] | [[3.5: Continuous Growth | Next Lesson]]
==Objectives:==
* Introduce (annual) compound interest
* Compound interest more frequently than annually
* Understand the difference between nominal and effective interest rate

==Important Items==
===Definitions:===
compound interest, nominal rate, effective rate
----

==Lesson Guide==

===Warm-Up===

Begin by asking your students if any of them have a bank account, and maybe ask if they know what interest rate they earn. In this section, we will explore different types of interest rates, and how they are computed (and how knowing which is which could help you make better investments!)

Have students do Problems 1 and 2.

You should expect that student will still be confused by effective annual percent rate. Make sure to look up the term ahead of time so that you can give them an accurate definition when they ask.

===Introduce (Annual) Compound Interest===

{{Ex-begin}}

Begin with an example of a bank with an interest rate that is ''compounded annually''. Choose a principal of <math>P=1000</math> dollars, and an interest rate, <math>r=12</math> percent, and have students help to compute the value in the account <math>t</math> years later. A table like the following one may be useful:

{| class="wikitable"
|-
|Years after Initial Deposit
|0
|1
|2
|3
|-
|Computation of Value
| 1000
| 1000(1.12)
| 1000(1.12)(1.12)
| 1000(1.12)(1.12)(1.12)
|-
|Value of Account
| 1000
| 1120
| 1254.4
| 1404.93
|}

Ask students what type of growth this account has, and then ask them to find a formula to describe the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit: <math></math>A(t)=P(1+r)^t.<math></math>

Stress that the bank is paying interest '''on the previously earned interest''' as well as on the principal each year.

{{Ex-end}}

For many instructors these are terms you may have not heard before. Make sure to look up these terms in the book so that you know what they are!

* The term [[compound interest]] refers to interest that is applied not only to the principal but also to previously earned interest.
* The [[nominal rate]] of an investment is the given interest rate, <math>r</math>.

*Ask students to identify the nominal rate in Problem 1.

===Compound Interest More Frequently than Annually===

{{Ex-begin}}

Begin with an example of a bank with an interest rate that is ''compounded monthly''. Choose a principal of <math>P=1000</math> dollars, and an annual interest rate, <math>r=12</math> percent, and have students help to compute the value in the account <math>t</math> years later. A table like the following one may be useful:

{| class="wikitable"
|-
|Months after Initial Deposit
|0
|1
|2
|3
|4
|5
|6
|7
|8
|9
|10
|11
|12
|-
|Computation of Value
| 1000
| 1000(1.01)
| 1000(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
| 1000(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)(1.01)
|-
|Value of Account
| 1000
| 1010
| 1020.1
| 1030.3
| 1040.6
| 1051.01
| 1061.52
| 1072.14
| 1082.86
| 1093.69
| 1104.62
| 1115.67
| 1126.83
|}

Emphasize to students that when a bank compounds interest monthly, for example, the bank will apply interest to the account, but it will be at a rate of <math>\frac{r}{12}</math>, '''NOT''' <math>r.</math> Describe that this is because they apply the interest rate of <math>r</math> in 12 increments. Fill out the following table:

{{Ex-end}}

Highlight that after 12 months, the account will have reached the amount it has after one total year of compounding. Ask the students if the amounts that the two accounts have at the end of one year are the same. Discuss how even with the same nominal rate, <math>r</math>, and principal, <math>P</math>, compounding at different frequencies has a big effect on the value of an account. Ask students to find a formula to describe the value <math>A(t)</math> of the account <math>t</math> years after the initial deposit:
<math>A(t)=P\left(1+\frac{r}{12}\right)^{12t}</math>

More generally, if a bank has an interest rate of <math>r</math> that compounds <math>n</math> times per year, the value <math>A(t)</math> of
the account <math>t</math> years after the initial deposit of the principal <math>P</math> is <math>A(t)=P\left(1+\frac{r}{n}\right)^{nt}.</math>
If you wish, you may refer to the above formula as the [[compound interest formula]].

*Have students do Problem 3.

*Discuss with your students how changing the frequency of compounding affects the value of the account. I.e., if Bank A and Bank B have the same interest rate <math>r</math> but Bank A compounds less frequently than Bank B, which bank yields more interest?

===Understand the difference between nominal and effective interest rate===

When the interest is compounded more frequently than once a year, the account effectively earns more than the nominal rate, and so we distinguish between the [[nominal rate]] and the [[effective rate]].

Suppose, for example, that an interest rate is 12\% compounded monthly (as in the example above). Explain that we refer to the 12\% as the [[nominal rate]]. When the interest is compounded more frequently than once a year, the account effectively earns more than the nominal rate, and so we distinguish between the nominal rate and the [[effective rate]]. The effective rate tells you how much interest the investment actually earns. This is sometimes called the APY (annual percentage yield) in the U.S.

{{Ex-begin}}
For each of the following two banks, determine what the effective interest rate is.

*'''Bank A''': Pays 12% interest compounded annually

Since an account paying 12% annual interest, compounded annually, grows by exactly 12% in one year, we have that the nominal rate is the same as its effective rate: both are 12\%.

*'''Bank B''': Pays 12% interest compounded monthly

The nominal rate is 12%. Using the compound interest formula, we know that after 12 months, our investment would be <math>1000(1.01)^{12}=1126.83</math>. The ''annual'' growth factor is
<math></math>\frac{1126.83}{1000}=1.12683.<math></math>
So, the account effectively earns 12.683\% interest in a year, so its effective interest rate is 12.683\%.

{{Ex-end}}

Remind students that <math>r</math> in the compound interest formula is the ''nominal rate'', not the effective rate.

The [[effective annual rate]] of an investment tells you how much interest the investment
actually earns per year. This is sometimes called the [[APY (annual percentage yield)]] in the U.S.

Make sure that students understand that if Bank A and Bank B have the same nominal interest rate, but Bank A compounds less frequently than Bank B, then the effective annual rate of Bank A will be less than the effective annual rate of Bank B.

Use Problem 3 on Worksheet 3.4 to compute the effective annual rate for each compounding frequency. Compare this with the nominal rate.

Have students do Problems 4-5.

Have students do Problems 6-7.

Remind students to refer to the compound interest rate formula (p. 157). Be sure to work your way around to each group. There are more problems listed here than most groups will finish in class, so you might want to encourage them to finish problems outside of class that their groups don't complete in class.

To bring closure to the above problems, as you work your way around the class, ask individuals to write the problems up on the board (you might find it best this time to check answers first so that students are sure about their answer going up on the board). Allow at least 3-5 minutes to go through the solutions together as a class. Depending on your time availability, go through a select few of the problems, and others only if you have time.

==Comments==

3.3: Graphs of Exponential Functions

2020-06-01T14:41:35Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Understand how changing the parameters <math>a</math..."

[[3.2: Comparing Exponential & Linear Growth | Prior Lesson]] | [[3.4: Compound Growth | Next Lesson]]

==Objectives:==

* Understand how changing the parameters <math>a</math> and <math>b</math> affects the shape of the graph of <math>Q(t)=a(b)^t</math>
* Review how to build exponential functions from word problems
* Recognize <math>y=0</math> is a horizontal asymptote for <math>Q(t)=a(b)^t</math>.

==Important Items==
===Definitions:===

horizontal asymptote

===Notes:===

We do not expect students to use limit notation fluently; they can also use the notation "as <math>t\to \infty</math>, <math>Q(t)\to 0</math>."
----

===Warm-Up===
Have students do Problems 1 and 2.

*[[Understand how changing the parameters <math>a</math> and <math>b</math> affects the shape of the graph of <math>Q(t)=a(b)^t</math>.]]

*We want to discuss how the values of <math>a</math> and <math>b</math> affect the graph of <math>Q(t)=a(b)^t</math>. First review some vocabulary. Point out that <math>b</math> is called the ''base'' of the exponential function (also the growth factor from \S3.1), and <math>a</math> is the initial value. Recall also that the growth rate is equal to <math>b-1</math>.

Write the following example on the board and give your students a minute to individually work out the solution before working it out on the board. Ask a student to help walk you through the solution.

{{Ex-begin}}

[[Q:]] If a quantity is modeled by <math>Q(t)=100(1.2)^t</math>, what is the growth rate?

[[A:]] The growth factor, <math>b</math>, is <math>1.2</math>. So <math>b-1=1.2-1=0.2</math>, or <math>20\%</math>, is the growth rate.

{{Ex-end}}

{{Ex-begin}}
Use Desmos to graph a "family" of exponential functions with a fixed growth factor, <math>b</math>, by varying <math>a</math>. Continue to play around with the values of <math>a</math> until students have convinced themselves of the pattern. Have students help you record their observations on the board.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/hukchtdgsk?embed
|width=500
|height=500
|border=0
}}

Next, use Desmos to graph a "family" of exponential functions with a fixed initial value, <math>a</math>, by varying <math>b</math>. Continue to play around with the values of <math>b</math> until students have convinced themselves of the pattern. Have students help you record their observations on the board. You may want to even graph multiple at the same time, and show how crossings can occur in both the first and second quadrant.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/wj6vlyx1je?embed
|width=500
|height=500
|border=0
}}

Alternatively, you can do this by graphing families of exponential functions by hand.

-Observations: If <math>Q(t)=a(b)^t</math>, then
* Changing <math>a</math> changes the <math>y</math>-intercept.
* The larger <math>b</math> is, the faster the graph grows.
* The smaller <math>b</math> is, the faster the graph decays.
* Two exponential functions must cross at some point if both their initial values are positive or both are negative and their bases are different.

{{Ex-end}}

*Have students do Problem 3.

*Use Problem 3 to talk about how we can compare exponential functions with different initial values '''and''' growth factors. In particular, talk about how we can determine whether or not they will cross in the first quadrant. I would avoid writing down "rules", but rather talk about how they can reason through the problem.

===Review how to build exponential functions from word problems===

*Do Problem 4 as an example of how to build exponential functions from word problems.
*Have students do Problems 5-7.

===Recognize <math>y=0</math> is a horizontal asymptote for <math>Q(t)=a(b)^t</math>.===

*Use one of the functions from Problems 5-7 to introduce the idea of end behavior. Ask, "''When we have exponential decay, what happens to <math>Q(t)</math> as <math>t</math> gets big?''" Discuss what you expect to happen to help students develop an intuitive picture of this function. You may want to sketch a graph of the function on the board. Test this hypothesis by constructing a table of values for large <math>t</math> (like the one below).

{| class="wikitable"
|-
|<math>t</math>
|1
|10
|50
|100
|-
|<math>Q(t)</math>
|
|
|
|
|}

Introduce limit notation to your students:
<math></math>\lim_{t\to \infty} Q(t)=0\qquad\text{-- or --}\qquad\text{as }t\to \infty\text{, }Q(t)\to 0.<math></math>

The horizontal line, the graph of <math>y=k</math>, is a [[horizontal asymptote]] of a function
<math>f(x)</math> if <math>\lim\limits_{x\to - \infty} f(x)=k \text{ or }\lim\limits_{x\to \infty} f(x)=k</math>

For exponential functions of the form <math>Q(t)=a(b)^t,</math> the
graph of <math>y=0</math> is a horizontal asymptote.
Indeed, one of the following will always be true:
<math>\text{as } t\to \infty,\;Q(t)\to 0 \qquad\text{--or--}\qquad\text{as } t\to - \infty, \;Q(t)\to 0.</math>

Graph two more exponential functions of the form <math>Q(t)=a(b)^t,</math> one that is growing and one that is decaying, to reinforce how one describes the "end behavior" of <math>Q(t)=a(b)^t</math> as the horizontal asymptote <math>y=0</math>.

*Have students do Problem 8.

*Go through Problem 9 with the students. This problem will help them tie some of what they have done in Chapter 3 with the materials from Chapter 2.

*Use Problem 10 to highlight the limitations of a model. We use exponential functions to model real world scenarios but there are places where they are just approximations. Knowing the limitations of a model is just as important as knowing the model.

3.2: Comparing Exponential & Linear Growth

2020-06-01T14:41:20Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Find exponential functions that fit the coordinates..."

[[ 3.1: Exponential Functions | Prior Lesson]] | [[3.3: Graphs of Exponential Functions | Next Lesson]]

==Objectives:==

*Find exponential functions that fit the coordinates of two points
*Understand the differences between linear and exponential functions
*Calculate ratios from a table of data to determine linear or exponential growth
*Determine whether a table of data can be represented using an exponential function (i.e., calculate ratios to see if there is a constant growth factor.)

----

==Important Items==

===Definitions:===

constant rate of change (reminder), constant growth factor

==Lesson Guide==

===Warm-Up===

Have students do Problems 1 and 2

===Find exponential functions that fit the coordinates of two points===

This section depends heavily on the graph of an exponential function. Graphing one or two exponential functions given a formula may be helpful if you have not done so yet. ''Now, given the graph of the function, how does one determine the corresponding function expression?''

Have students do Problem 3. After most of the students have completed Problem 3 ask your students which of these represents a ''linear function'' and which represents an ''exponential function''. Use this Problem to discuss/remind your students that linear functions have a [[constant rate of change]] (the difference of outputs is constant), whereas exponential functions have a [[constant growth factor]] (the quotient of outputs is constant).

Have students do Problem 4. Make sure that the students have checked every ratio. On exams, many students miss problems because they do not check every ratio or checked every point in the equation they have built

Do an example on the board that illustrates how to find the initial value of the exponential function, and how to use two points to algebraically solve for the growth factor. Emphasize the similarity in this process to how one can use the graph of a linear function to find its formula.

{{Ex-begin}}
Suppose we have a certain sample of bacteria that we know grows exponentially. Further suppose we know that 2 hours after placing 250 bacteria in a petri dish there were 1000 bacteria in the dish.

We know two points <math>P(0)=750</math>, and <math>P(2)=1000</math>. Then, <math>750=P_0(b)^0</math> which means that <math>P_0=750</math> since <math>b^0=1</math>. From here we can say

<math>
1000 = 750 (b)^2
</math>

<math>
\frac{4}{3} = b^2
</math>

<math>
\frac{2}{\sqrt{3}} =b
</math>

Therefore, we can model the situation using the equation <math>P(t)=750\left(\frac{2}{\sqrt{3}} \right)^t</math>
{{Ex-end}}

Have students do Problem 5. Students will find the last one particularly challenging; let them try it first and then discuss as a class that we can plug both points into the general form of an exponential equation and then substitute one into the other.

===Understand the differences between linear and exponential functions===

Reiterate the following with the class: Linear functions represent quantities with a constant rate of change, whereas [[exponential functions represent quantities that change at a constant growth factor]]. Stress that the difference of successive outputs of a linear function is constant, the quotient of successive outputs of an exponential functions is constant. You may also want to point out that linear functions are [[adding a fixed amount at each time interval]], whereas exponential functions are [[multiplying by a fixed amount at each time interval]]. Use the following example to demonstrate these concepts.

{{Ex-begin}}
Which is linear and which is exponential? If linear, what is the constant rate of change? If exponential, what is the growth factor?
*A population doubles each year.
*The cost of a utility bill increases by 10 dollars each year.

Discuss/write on board:
*Exponential with a constant growth factor of 2;
*Linear with a constant rate of change 10.

{{Ex-end}}

Have students do Problem 6-7.

Use Problem 6 to discuss how linear functions have a [[constant rate of change]] (the difference of outputs is constant), whereas exponential functions have a [[constant growth factor]] (the quotient of outputs is constant).

===Calculate ratios from a table of data to determine linear or exponential growth===
{{Ex-begin}}
On the board, write a completed table such as the following where <math>f(t)</math> is linear, <math>g(t)</math> is exponential, and <math>h(t)</math>
is neither linear nor exponential. Show students how to compare pairs of points to determine whether or not the table
could represent a linear or exponential function.

{| class="wikitable"
|-
|<math>t</math>
|0
|1
|2
|3
|-
|<math>f(t)</math>
|
|
|
|
|-
|<math>g(t)</math>
|
|
|
|
|-
|<math>h(t)</math>
|
|
|
|
|}
{{Ex-end}}

Have students do Problem 8.

Again emphasize that linear functions have a constant rate of change and exponential functions have a constant growth factor. Comparing them by talking about how linear functions grow/decay via addition while exponential functions grow/decay via multiplication is important to highlight.

Have students work Problem 9.

==Comments==
----
For the pedagogy project, a slightly edited version of this lesson plan can be here:
http://www.math.unl.edu/~nwakefield2/FYM/index.php/3.2:_Comparing_Exponential_%26_Linear_Growth/DanaLacey

3.1: Exponential Functions

2020-06-01T14:41:08Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Recognize when..."

[[ 2.2: A Brief Introduction to Composite & Inverse Functions | Prior Lesson]] | [[3.2: Comparing Exponential & Linear Growth | Next Lesson]]

==Objectives:==
*Recognize when a function is exponential
*Understand exponential growth and decay
*Build exponential equations
*Evaluate and interpret exponential functions
*Determine the growth rate from the growth factor, and vice versa, for a given exponential model.
*Create an equation for an exponential function to represent a quantity's growth or decay.
*Find the value of an exponential function at a given time <math>t</math>.

----
==Important Items==

===Definitions:===

exponential function, growth factor, growth rate

===Notes:===
While the old textbook used the notation <math>Q=a(b)^t</math>, we use function notation <math>Q(t)=a(b)^t</math> or <math>f(t)=a(b)^t</math> whenever possible. We want to emphasize functional notation. Furthermore, try to always use parentheses to denote multiplication, i.e., <math>f(t)=a(b)^t</math>, rather than <math>f(t)=a\cdot b^t</math> or <math>f(t)=ab^t</math>. This makes it easier to transition to talking about the growth factor <math>b</math> in <math>f(t)=a(b)^t</math> and the annual growth rate <math>r</math> in <math>f(t)=a(1+r)^t</math>.

The two terms that you want to emphasize are growth rate and growth factor. Be aware that some textbooks will use other terms as well in the same contexts; for instance, the book would refer to "percentage growth rate," when they just mean the percent form of growth rate. To avoid confusion, for the most part you can ignore the "percentage" and just refer to growth rate/factor.

==Lesson Guide==

===Warm-Up===
Have students do Problem 1

===Recognize when a function is exponential===

Use an example to explore the notion of an exponential function. Your example should help students identify exponential functions and distinguish them from linear functions.

{{Ex-begin}}
You deposit 500 dollars into an account that earns 4.5% interest annually. Make a table showing the value of the account 0, 1, 2, 3, and <math>t</math> years after the money was originally deposited.
 

{| class="wikitable"
|-
|Years (<math>t</math>)
|0
|1
|2
|3
|...
|t
|-
|<math>A(t)</math>= amount after <math>t</math> years
|500
|500+500(0.045)=500(1.045)=522.50
|500(1.045)+500(1.045)(0.045)=500(1.045)^2=546.01
|500(1.045)^2+500(1.045)^2(0.045)=500(1.045)^3=570.58
|
|500(1.045)^t
|}

Use the table to discuss the ''ratio of successive outputs'' of <math>A(t)</math>; the ratio of successive outputs is the growth factor. Note that it is ''constant'' (the ratio is the same in each case): <math>1.045</math> (not <math>0.045</math>).

Note: Students may be uncomfortable with the term "ratio" so you might call it "the amount we multiply each time to get to the next output." It is informal, but will probably make more sense to students.

Graph the function you found on the board, using values from the table and confirm your graph with a graphing calculator. Discuss graph features like intercepts, increasing, domain and range, etc.

{{Ex-end}}

Definition: An [[exponential function]] is a function of the form <math>Q(t)=a(b)^t,</math> (<math>a\not=0</math>, <math>b>0</math>, <math>b\neq1</math>),
where <math>a</math> is the initial value of <math>Q(t)</math> (at <math>t=0</math>) and <math>b</math>, the base,
is the [[growth factor]]. The [[growth rate]] is <math>b-1</math>.

Have students do Problem 2.

Ask students what you must check in order to determine whether a function is exponential. Guide them to telling you that it must have a constant growth factor. Then have them work on Problem 3. Have each table write their final answer to number 3 on the board. The goal is to emphasize the following:

Notice that they might get different answers if they use an exponential formula or if they multiply by 1.25 and round to the nearest whatever each time. If this comes up, you might discuss how rounding off can make errors worse and worse as time goes on (even though we are talking about discrete items here.) Encourage students to use an exponential formula rather than calculating the amount of hats year by year.

===Understand exponential growth and decay===

Ask students when you have exponential growth vs. exponential decay:

Suppose <math>a>0</math>.

*If there is exponential growth, then <math>b>1</math>.
*If there is exponential decay, then <math>0<b<1</math>.

Use Problem 4 to explain examples of an exponential function with growth and another with decay. Sketching the graphs of these two examples is a good idea. Finally, have the students complete Problem 4 in their groups.

{{Ex-begin}}
The grades of six students are given in problem 4. Let's look at the graphs of the equations given by <math>P(t)=97(1.001)^t</math> and <math>P(t)=85(0.89)^t</math>.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/6nnexlbnyx?embed
|width=500
|height=500
|border=0
}}
{{#widget:Iframe
|url=https://www.desmos.com/calculator/rxdr48xtze?embed
|width=500
|height=500
|border=0
}}

{{Ex-end}}

===Build exponential equations===

Tie together the examples in Problem 5 to illustrate how one might build an exponential equation from a word problem. You might have students work on 5 for a few minutes and then go through the problem as an entire class.

Highlight the relationship between the growth factor <math>b</math> and growth rate in your example, and be sure to write it down formally:

Growth Factor: <math>b</math>
Growth Rate: <math>b-1</math>

Have students do Problems 6-7.

===Evaluate and interpret exponential functions===

Have students do Problems 8-9and then go over one of these making sure to emphasize units and interpretation.

2.2: A Brief Introduction to Composite & Inverse Functions

2020-06-01T14:40:53Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives== *Understand function composition (notation, evaluation, and in word..."

[[ 2.1 Piecewise Functions | Prior Lesson]] | [[ 3.1: Exponential Functions | Next Lesson]]

==Objectives==
*Understand function composition (notation, evaluation, and in word problems)
*Understand how to find inverse functions and interpret their inputs and outputs

----

==Lesson Guide==

This section is devoted to a ''brief'' introduction to the concept and interpretation of an inverse function. The algebra of finding an inverse and a more in-depth discussion is saved for later, so DO NOT get bogged down in algebraically finding inverse functions at this time. You are strongly encouraged to use the function diagram and bubble diagrams in this section that that theme will come up often throughout the course. There are some webwork problems that involve finding the inverse of a linear function. I find this easiest to cover using a function diagram and talking about how an inverse function goes the opposite direction and so undoes the order of operations.

The notation is introduced here so it can be used when we talk about logs. Be certain to stress that the notation <math>f^{-1} (x)</math> does not mean <math>\frac{1}{f(x)}</math>! At this time, the students need to realize that with the notation <math>f^{-1}</math>, the input and output are switched from those of <math>f</math>. (For now, try to avoid using <math>f(x)</math> and <math>f^{-1}(x)</math>.) It is much easier for students to understand <math>Q=f(P)</math> and <math>P=f^{-1} (Q)</math>. Using the same ``name" for the input of <math>f</math> and <math>f^{-1}</math> can be quite confusing for students, especially since we try to focus on understanding that the ``input and output are switched".)

Many students have encountered inverses before, but it is unlikely that they connect practical meaning with calculating inverses. They may know some procedure like ``swap x and y, and then solve for <math>y</math>," but have no understanding of what they are doing. In Section 2.4 we stress function notation - NOT the procedure for finding general inverse formulas.

----

====Warm-Up====

Have students do Problems 1 and 2.

We have not really talked about how to evaluate <math>f(f(4))</math> yet, but you should be able to help students work through it.

====Understand function composition (notation, evaluation, and in word problems)====

Introduce the idea of composing two functions using a diagram:
<math>x\xrightarrow{f} f(x)\xrightarrow{g} g(f(x))</math>
Given an input <math>x</math> for a function <math>f</math>, we get an output <math>y=f(x).</math> So long as <math>y=f(x)</math> is an allowed input for <math>g,</math> i.e., <math>y=f(x)</math> is in the domain of <math>g</math>, we can evaluate <math>g(y).</math> This final output or ``composition" is written as <math>g(f(x)).</math>

Do an example of evaluating the composition of two given function formulas at a given input. Use the arrow diagram above with your specific example. Eventually, you do many examples without the diagram but students need to have a conceptual understanding using the arrow diagram.

{{Ex-begin}}
Suppose that <math>f(x)=x^2+2</math> and <math>g(x)=3x+1</math>. Calculate <math>f(g(2))</math>.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/bd6bhpzfbc?embed
|width=500
|height=500
|border=0
}}
{{Ex-end}}

Have students do Problems 3-5.

Some of the Webwork problems will ask students to compose two functions that are represented in various forms: as graphs, in a table, ordered pairs. To prepare students for this, it would be helpful to look at these Webwork problems beforehand and present something similar to the following example in class.

{{Ex-begin}}
Using the table and graph below, evaluate the following quantities.

*<math>h(2)</math> where <math>h(x)=g(f(x))</math>
*<math>h(0)</math> where <math>h(x)=f(f(x))</math>
*<math>h(4)</math> where <math>h(x)=g(x^2)</math>

{{#widget:Iframe
|url=https://www.desmos.com/calculator/1vcmjj5rb8?embed
|width=500
|height=500
|border=0
}}

{| class="wikitable"
|-
|<math>x</math>
|0
|1
|2
|3
|4
|-
|<math>f(x)</math>
|2
|0
|5
| -5
|1
|}

{{Ex-end}}

===Understand how to find inverse functions and interpret their inputs and outputs===

Write the following notation on the board, don't get caught in too many details yet. You will have students work through the details by doing problems 6 and 7.

Given a function <math>y=f(x)</math>, the inverse function will have ''input'' <math>y</math> and ''output'' <math>x</math>:<math>x=f^{-1} (y),</math> i.e.,
the domain of <math>f</math> is the range of <math>f^{-1}</math> and the range of <math>f</math> is the domain of <math>f^{-1}.</math> We call <math>f^{-1}</math>
the [[inverse function]] of <math>f.</math>

Warning: In <math>f^{-1}</math> the <math>-1</math> is NOT an exponent. '''<math>f^{-1}(x) \not= \frac{1}{f(x)}</math>'''

Have students do problems 6 and 7.

Have groups present on problems 6 and 7. After students have presented use their work to talk about how you might identify the domain and range of the inverse <math>f^{-1}</math> given a function <math>f</math>.

Have students do problem 8.

On problem 8 they have to find the inverse functions. You should be prepared to talk students through the process.

Have students do the Synthesis Problem (Problem 9) when they finish the rest.

2.1 Piecewise Functions

2020-06-01T14:40:40Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== * Evaluate piecewise functions * Graph p..."

[[ 1.7 Domain & Range | Prior Lesson]] | [[2.2: A Brief Introduction to Composite & Inverse Functions | Next Lesson]]

==Objectives:==
* Evaluate piecewise functions
* Graph piecewise functions
* Create piecewise functions from a word problem or a graph

===Definitions:===

----

===Note:===

We begin with the Interlude: Introduction to Piecewise Functions. Piecewise functions often prove troubling to students throughout the semester. Being especially thorough with the introductory material should be helpful.

==Lesson Guide==

===Warm-Up===

Have students do the interlude and Problems 1 and 2. If you like, you can go through it together as a class or show them a similar example first. Alternatively, you can just have them work their way through the problems on their own and then discuss as a class what piecewise functions are. Many of your students will report that they don't know how to graph the function. If this is the case then walk them through plotting points and connecting those points with reasonable curves.

{{Ex-begin}}
Students will be uncomfortable with piecewise function notation. Take a few minutes to interpret what this seemingly strange notation means. One way to do this is the following:

Let <math>g(x)</math> be the function that is <math>x^2-2</math> when <math>x<1</math> and is <math>x+2</math> when <math>x\geq 1</math>. We might write this as

<math display="block">

g(x) = x^2-2 \text{ if } x<1 \text{, and}</math><math>
g(x) = x+2 \text{ if } x\geq 1.
</math>

To save space (and be lazier...) we use a curly bracket:
<math>
f(x)= \begin{cases}
x^2 & \text{ for } x< 1 \\
x+2 & \text{ for } 1\leq x
\end{cases}
</math>

'''Note:''' We may use the words ``for," ``if," ``when," or nothing at all before writing the domain of each piece.
}}

'''Note:''' Students may also struggle with the last part of the interlude, which has them evaluate a piecewise function at specific <math>x</math>-values. Make sure to emphasize to students that the new piecewise function, <math>g(x)</math>, is what we built out of the two pieces. When <math>x<1</math>, we use the rule given by the first line of the piecewise function, and when <math>x \geq 1</math>, we use the second line.

'''Optional:''' Do this process as a physical demonstration. Come in with graphs for <math>y=x^2</math> and <math>y=x+2</math> on separate sheets of paper (each piece could be a different color). Make sure the scales of each of the graphs match. Then physically cut each graph and paste them together correctly. Be sure to actually draw the final answer with an open and closed circle separately on the board with open and closed circles (students can then copy the finished version into their notes). The doc cam would be a useful tool in this case.

{{Ex-end}}

===Evaluate piecewise functions===

Demonstrate how to evaluate a piecewise function at a given input.

{{Ex-begin}}
Again, let <math>g(x)</math> be the function that is <math>x^2-2</math> when <math>x<1</math> and is <math>x+2</math> when <math>x\geq 1</math>. Then

<math>
g(x)= \begin{cases}
x^2-2 & \text{ for } x< 1 \\
x+2 & \text{ for } 1\leq x
\end{cases}.
</math>

Someone might as you to evaluate <math>g(0)</math>. In order to do this evaluation we note that <math>0 < 1</math> hence we need to plug zero into the equation <math>x^2-2</math>. Therefore, <math>g(0)=0^2-2=-2</math>. Similarly, <math>g(2)=2+2</math>.

{{Ex-end}}

Have students do Problem 3.

Highlight the domain of each of the functions and also carefully explain what the open circle and closed circle notation means on the graph of these functions (i.e., how this corresponds to the domains of the function pieces).

===Graph piecewise functions===

Have students do Problems 4 and 5. Be sure to tell students that there are places on Problem 5 where the function is not even defined.

===Create piecewise functions from a word problem or a graph===

Ask students to try and solve problems 6-8. Students may resist at first but there should not be any issue with letting them try their hand at the Problems without an explicit example first.

Have students do Problems 6-8.

'''Note:''' Problem 7 has some major issues. However, we have chosen not to fix problem 7 because we think these issues can be instructive if you know about them ahead of time. In particular, the ticket function is not defined for values between 74 mph and 75 mph. Make sure that you point this out to your students and use this to explain that sometimes in the real world people say things one way, but really mean something different. Ask students how they might fix the problem and use this as an opportunity to talk about where this problems breaks down.

Groups that are moving along well can continue by working on Problems 9 and 10 but not everybody will get this far. Tell students that the important thing is that they get through Problem 8.

1.7 Domain & Range

2020-06-01T14:40:29Z

Nwakefield2: Created page with "Prior Lesson | Next Lesson ==Objectives:== * Identify the domain and range of a function represented in..."

[[1.6 Function Notation Input & Output|Prior Lesson]] | [[2.1 Piecewise Functions | Next Lesson]]

==Objectives:==
* Identify the domain and range of a function represented in various forms

==Important Items==
===Definitions:===
domain, range

===Notes:===
We often will model discrete situations using continuous functions. We will {\em not} be covering finding the domain of a discrete situation (although there are some examples of this in the text). Tell students that unless a problem asks for a specific format of notation, it is acceptable to use whichever is most familiar, i.e., the domain of <math>f(x)=\sqrt{x}</math> is any of: <math>0\leq x</math>, <math>\{x:x\geq 0\}</math>, or <math>[0, \infty)</math>.

----

==Lesson Guide==

===Warm-Up===

Have students do Problems 1 and 2.

Have students work on problem 3. You can use this problem to lead into your discussion of domain and range.

===Identify the domain and range of a function represented in various forms===

If <math>y=f(x)</math> is a function, then
* The [[domain]] of <math>f</math> is the set of input values, <math>x</math>, which yield an output value.
* The [[range]] of <math>f</math> is the corresponding set of output values, <math>y</math>.

Give students multiple ways to think about the domain and range; i.e., the domain is the ``set of allowed inputs.'' Additionally, you may graph a function in black and use a different color to trace along the <math>x</math>-axis to denote the domain and yet another color to trace along the <math>y</math>-axis to denote the range.

Have students do Problem 4.

Do several examples similar to Problems 4-6. Examples should show different ways in which one can determine the domain and range (use a table, ordered pairs, a graph, a formula, and a word problem). Also be sure to give an example of when something is NOT in the domain or NOT in the range of a given function. It might be helpful to use bubble diagrams of the functions here. e.g.
{{Ex-begin}}
'''Find the domain of <math>f(x)=\sqrt{x+8}</math>.'''

{{#widget:Iframe
|url=https://www.desmos.com/calculator/rimlmjyqkw?embed
|width=750
|height=750
|border=0
}}

{{Ex-end}}

{{Ex-begin}}
'''Find the range of <math>f(x)=\sqrt{x+8}</math>.'''

{{#widget:Iframe
|url=https://www.desmos.com/calculator/4balokqjie?embed
|width=750
|height=750
|border=0
}}

From the graph we can see the the range is <math>([0,\infty)</math>.

{{Ex-end}}

Have students do Problems 5-11.
This section is consistently difficult for students. Throughout class, asking students to present their solutions on the board and also giving more examples as questions arise will be helpful.

You should also do an example of how to find the range of a function if you specify a domain for a function, because this appears a lot on the homework. Something linear may work nicely.

==Comments==
----
* Problem 9 should reference Problem 4 rather than Problem 3.
* The spacing on Problem 3 should be fixed for the next version of the course packet (misplaced \newpage command)

1.6 Function Notation Input & Output

2020-06-01T14:40:16Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Interpret inputs and outputs of a function *Evaluate a function at..."

[[1.5 Comparing Linear Functions | Prior Lesson]] | [[1.7 Domain & Range | Next Lesson]]

==Objectives:==

*Interpret inputs and outputs of a function
*Evaluate a function at a given input and solve a function equation with a given output
*Extend objectives (i) and (ii) to graphs of functions

;Definitions: There are no major definitions in this section.

----
==Lesson Guide==

(Warm-Up) Have students do Problems 1 and 2.

Note: Students may not actually remember how to solve <math>H(x)=9</math> in Problem 2. Don't get too bogged down on that here.

===Interpret inputs and outputs of a function===

Have students spend about 10 minutes working on Problem 3 in their groups. This problem incorporates a function given in words that does not have a formula. You may want to remind your students that this is still a valid function, and you will certainly want to model correct language for them. As you circle the room make sure to interrupt the class and:
*remind students of how to determine what objects are the inputs or outputs of a function,
*remind students that f(input) is an output value,
*emphasize writing the units given in the Problem and a complete sentence, and
*get students comfortable with evaluating a function at a given input.

Use the document camera or the whiteboard and have a group present their answers to Problem 3.

Some questions to ask students in your discussion include: Is <math>f(103)</math> a function or a number? What does <math>f(15)=73</math> mean on a graph? Would we expect this function to be increasing or decreasing?
(You may want to mention to your students that we will ask them to interpret functions in complete sentences on exams.)

===Evaluate a function at a given input and solve a function equation with a given output===

Often we are given formulas for functions and asked to evaluate at a given input or solve for a given output. Do an example like Problem 4 to illustrate the difference between evaluating and solving.

{{Ex-begin}}
[[File:Screen Shot 2019-12-05 at 3.53.05 PM.png|thumb]]
{{Ex-end}}

Note:Contrast language here with previous example. That is emphasize that we're using words like ``find" and ``solve". Make sure your input arrows point towards the number in parentheses and not to the entire expression <math>c(\frac{1}{2})</math>. You may also note that <math>(\frac{1}{2})^2-3(\frac{1}{2})</math> is another way of writing the output.

*Remind students how to evaluate a function at a given input.
*Given an output <math>y</math>, emphasize to students that solving the equation <math>f(x)=y</math> is not the same as plugging the value <math>y</math> into the function as an input.

Have the students work on Problem 4

Have students do Problem 5.

===Extend objectives (i) and (ii) to graphs of functions===

Have students do Problem 6.

In part (a) students may struggle with the idea that the output is the velocity. Note in part (b) that we don't know for sure, but it's probably 8 minutes, as he stops suddenly and then walks back the way he came. On part (c), students may be tempted to say intervals where he is walking at 3mph, but his fastest speed is actually 4mph. On (d), students may give points rather than intervals for their answers (e.g. "t=1 and t=4" instead of "[1,4]" or "1<t<4")

Once students have worked for a while lead a whole class discussion on the problem.

Have students do Problem 7 and 8. If time permits you should have a group present their answer for Problem 8

==Comments==
----
Some hw problems involve solving for equations like (x+1)/(x+3), and composing a funciton g(x) with 1/(x+2). I recommend doing an example like these so they have them for reference. (This will not be an issue for Fall 2019 onward.)

1.5 Comparing Linear Functions

2020-06-01T14:40:03Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Compare the graphs of linear functions with differen..."

[[1.4 Finding Linear Functions | Prior Lesson]] | [[1.6 Function Notation Input & Output | Next Lesson]]

==Objectives:==
*Compare the graphs of linear functions with different slopes and y-intercepts
*Given a line, find the slopes of a parallel line and a perpendicular line
*Write equations for vertical and horizontal lines

----
;Definitions: perpendicular, parallel, same line

----

==Lesson Guide==

===Warm-Up===
Have students do Problems 1 and 2.

One way to approach this activity is to have students try to work through it on their own for 5 minutes, and then spend 10 minutes as a class going through the graph. As you go through the questions collaborate with the students to highlight the following facts:

* Decreasing lines have a negative slope and increasing lines have a positive slope. Remember, we ``read" a graph from left to right. So <math>C</math>, <math>D</math> and <math>E</math> are increasing functions, <math>B</math> is a decreasing function and <math>A</math> is neither increasing nor decreasing.
*The ``steepness" of a graph comes from the magnitude, not the ``positiveness" of the slope. So <math>B: y = 2 -x</math> and <math>D: y = x-2</math> have the same steepness even though one has negative and the other positive slope.
*In fact, <math>B</math> and <math>D</math> are perpendicular lines. How do we know? We can tell because <math></math>\text{slope}(D) \times \text{slope}(B) = 1 \times -1= -1<math></math> Note that students (and the text) may also use the term "negative reciprocals." Point out that these are equivalent.
*Horizontal lines have a slope of zero. So for horizontal lines <math>y = mx + b</math> becomes <math>y= b</math>. The equation for <math>A</math> is therefore <math>y = 5</math>
*The <math>y</math>-intercept is just the point where the graph crosses the <math>y</math>-axis. What are the <math>y</math>-intercepts for <math>A</math>, <math>B</math>, <math>D</math> and <math>E</math>? <math>A: (0,5)</math>, <math>B: (0,2)</math>, <math>D \& E: (0,-2)</math>.
'''You might even choose to write some of these facts on the board!''' If possible it is good to let students make the observations and then write them on the board as students list these facts. If students can create the table with your prodding they will be much better off.

====Understand how changing the slope and <math>y</math>-intercept changes the graph of a linear equation====

Use Problem 1 to discuss how the slope reflects whether the graph of the linear function is increasing or decreasing.

----
If (a function defined by) <math>y=mx+b</math> is an equation for a line with slope <math>m</math> and <math>y</math>-intercept <math>(0,b)</math>, then
*<math>m>0</math> means that the function is '''increasing''',
*<math>m<0</math> means that the function is '''decreasing''',
*<math>m=0</math> means that the graph of the function is the horizontal line with equation <math>y=b</math>, and
*the larger the magnitude of <math>m</math>, the steeper its graph is.
----

====Understand the relationship between slopes of parallel and perpendicular lines====

This section focuses on the relationship between two lines, specifically the relationship between their slopes. Ask students the various relationships that two lines could have (they intersect at a point, they never intersect, or they are the same line). Give a few examples of graphs of pairs of lines and ask students what the relationship is between the lines.

{{Ex-begin}}
{{#widget:Iframe
|url=https://www.desmos.com/calculator/nlq1hp7irj?embed
|width=500
|height=500
|border=0
}}

{{#widget:Iframe
|url=https://www.desmos.com/calculator/ewtpfrrn4b?embed
|width=500
|height=500
|border=0
}}

{{#widget:Iframe
|url=https://www.desmos.com/calculator/5hvl6jmogw?embed
|width=500
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}}
{{Ex-end}}

Once you have done the examples, ask how we could determine the relationship by simply looking at the linear equations of the lines.
----
Let line 1 be given by the formula <math>y = m_1x + b_1</math> and let line 2 be given by the formula <math>y = m_2x + b_2</math>. Then, we say that
*lines 1 and 2 are \underline{perpendicular} if <math>m_1 = \frac{-1}{m_2}</math> (negative reciprocal).
*lines 1 and 2 are \underline{parallel} if <math>m_1 = m_2</math> and <math>b_1\neq b_2</math>. (Note: some homework problems may require that <math>b_1\neq b_2</math> but for exams we will make a point that <math>m_1 = m_2</math>. There are two camps on whether it is necessary that <math>b_1\neq b_2</math> and we will try to avoid taking a side.)
*lines 1 and 2 are the \underline{same line} if <math>m_1=m_2</math> {\em and} <math>b_1=b_2.</math>
----

The statement <math>m_1 = \frac{-1}{m_2}</math> may confuse students, so saying ``negative reciprocal" may actually be more helpful to them.

Have students do Problem 3(a-g).

====Write equations for vertical and horizontal lines====

Remind students that vertical lines have a fixed <math>x</math>-coordinate for every point on the line. So, we describe a vertical line by simply writing <math>x=c</math> for whatever that constant <math>x</math>-coordinate <math>c</math> is.
----
For any constant <math>k</math>:

*The graph of the equation <math>y=k</math> is a horizontal line through <math>(0,k)</math> and its slope is zero.

*The graph of the equation <math>x=k</math> is a vertical line through <math>(k,0)</math> and its slope is undefined.

Have students finish Problem 3.

Tell students that Problem 4 has shown up on many of the exams and they should make sure they have that mastered.

Have students finish Problem 5. Use a graph to show students how the word problem and graph are tied together.

==Comments==

*Students have not actually been taught the meaning of the word magnitude in 100A, so it would probably be good to define magnitude for them.

*I think problem 3 should really be split up. Even in the lesson plan, we don't do the whole problem at once, so I don't see a reason why it should all be one problem. It's just a lot for students to look at and could be overwhelming.

*Regarding #3, I like to assign each group 2-3 parts to do and write up on the board. That way it's not as overwhelming. -Juliana

I have found that stating that perpendicular lines have the negative reciprocal of each other has usually confused students more. It has often helped to actually to present m_1 times m_2 = -1. And then have them solve the equation. Last semester most students forgot the negative with the reciprocal.

1.4 Finding Linear Functions

2020-06-01T14:39:12Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Review and use the slope-intercept and point-slope forms for a lin..."

[[1.3 Linear Functions | Prior Lesson]] | [[1.5 Comparing Linear Functions | Next Lesson]]

==Objectives:==
*Review and use the slope-intercept and point-slope forms for a line
*Use different forms of a line to find the formula of a linear function that is represented by a table, graph, set of points, etc.
;Definitions: slope-intercept form, point-slope form

----

==Lesson Guide==
Have students do Problems 1 and 2.

*Once most of the students have finished problems 1 and 2 you should give a very short statement to the whole class to tie the warm-up together with the rest of what you will be covering. Something like "In problem 1 and problem 2 we used one important characteristic of a linear equation, what was that characteristic?" Allow students to share answers then ask "what is the other component of a linear equation?" Wait for responses. "Now I want you to use this to help you solve problem 3."

==Review and use the slope-intercept and point-slope forms for a linear equation==
Have students do Problem 3.

This may feel odd, why are we having students find the formula when we have not shown them how to do so. The vast majority of students actually know how to find the formula. We are pushing them to discover a formula on their own. Pay attention to what they do and use their ideas to help guide your lecture for the rest of the day.

Ask specific students to provide their solutions and how they got them. Ask for various methods students used on these Problems until someone offers ``slope-intercept form" and ``point-slope form" as strategies.

----
The [[slope-intercept form]] of a linear equation with slope <math>m</math> and <math>y</math>-intercept <math>(0,b)</math> is
<math>
y = mx+b.
</math>
A [[point-slope form]] of a linear equation with slope <math>m</math> and a point <math>(x_0,y_0)</math> on the line is
<math>
y - y_0 = m(x-x_0)
</math>
----

==Use different forms of a line to find the formula of a linear function that is represented by a table, graph, set of points, etc.==

Have students do Problems 4-7.

Have students write their answers on the board and make sure that the answers cover examples where you find the formula for a linear function given a graph, a set of points, or a word problem. You may want to show how to do such a problem given a table. In particular, you could go back to previous examples in the course packet (\S1.3) and use these tables and graphs now.

Once students have mastered the more basic Problems (5-7) have them move onto Problems 8. Be careful not to give away too much information early on in the semester.

End class by having a student present their answer to Problem 8. Tell students to talk about Problem 8 and make sure everyone at their table can master Problem 8. Problem 8 often comes up on either quizzes or exams.

Remember
*Students may need some reminder that they can find the slope using the formula.
*Students will likely struggle with finding the slope. Just remind them that they should know the formula for slope as well as the "rise over run" notion for slope, especially on the second graph.
*Students may have a difficult time determining which is the <math>x</math>-value and which is the <math>y</math>-value.
*Students will likely struggle with the idea that they need to solve for the unknown values.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

1.3 Linear Functions

2020-06-01T14:38:59Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Determine when a function might be linear from a sample of points *Ide..."

[[1.2 Rate of Change | Prior Lesson]] | [[1.4 Finding Linear Functions | Next Lesson]]

==Objectives:==
*Determine when a function might be linear from a sample of points
*Identify the slope, x-intercept, and y-intercept of a linear function
*Explain the significance of the slope, x-intercept, and y-intercept of a linear function modeling a word problem
*Create linear functions from word problems

----

;Definitions: linear function, slope, <math>x</math>-intercept, <math>y</math>-intercept

----

==Lesson Guide==

===Warm-Up===

Have students do Problems 1.

===Determine when a function is linear from a sample of points===
Recall some of the problems in the 1.2 worksheet with a non-constant rate of change. Discuss with your students that linear functions always have a ''constant rate of change''.

The average rate of change of a function is usually different on different intervals. This is why linear functions are so very special. What is the definition of a linear function? Why do we call it linear? Coax the two essential points out of your students. Write them on the board.

----
A '''linear function''' is a function with a constant rate of change. The graph of a linear function is a line.
----

Do an example where you check if a table of points that represent a function can be linear. Explain to students that linear functions should have a constant rate of change, i.e., if we compute the average rate of change between any two pairs of points, it should always be the same. Make a point that in order to confirm a function is linear you must check every point.

{{Ex-begin}}

Draw the graph. Circle the <math>x</math>- and <math>y</math>- intercepts and ask the students if they know the names for these points. Label the points accordingly. Remind the students that the <math>y</math>-intercept occurs when the input is 0; an <math>x</math>- intercept occurs when the output is zero. We can also find the slope of the linear function by calculating the rate of change on any interval. Finally, with the initial value, <math>f(0)</math>, and the slope we can write an equation for any linear function.

''It is important to drill home how to find the <math>x</math>- and <math>y</math>-intercepts as this will be used over and over again later in the course.''

Write the following on the board and then draw the following graph.
*A linear function is a function with a constant rate of change
*The graph of a linear function is a line

[[File:Linearfunction.png|thumb]]

Output = Initial Value + Rate of Change <math>\times</math> Input

<math>f(x) = b + mx</math>

We can calculate <math>m</math> by the following formula: If <math>(x_0,y_0)</math> and <math>(x_1,y_1)</math> are two points on a line, the slope <math>m</math> is given by

<math>
m = \frac{\text{Rise}}{\text{Run}} = \frac{\Delta y}{\Delta x} = \frac{y_1-y_0}{x_1-x_0}.
</math>

In general, to find the <math>y</math>-intercept of a function, <math>f(x)</math>, we evaluate <math>f(0)</math>. In the case <math>f(x) = mx+b</math>, we see that <math>f(0) = b</math>, so <math>(0,b)</math> is the <math>y</math>-intercept (also called the initial value).

To find the <math>x</math> intercept, we solve for <math>x</math> in <math>f(x) = 0</math>. In our case, we set <math>mx+b = 0</math>, and we see that <math>x = -\frac{b}{m}</math>.

{{Ex-end}}

Have students do Problem 3.

===Understand the terms: slope, <math>x</math>-intercept, <math>y</math>-intercept ===

Graph a linear function using a formula obtained from a table in Problem 3 that could represent a linear function. Identify the slope, the <math>x</math>-intercept, and the <math>y</math>-intercept. Discuss what these mean and write it formally on the board:

----
Given a linear function <math>y=f(x),</math>
* the \underline{slope} is the the \textbf{constant} rate of change, and can be computed as <math>\frac{f(x_1)-f(x_2)}{x_1-x_2}</math> given any distinct <math>x</math> values <math>x_1</math> and <math>x_2.</math>
*The \underline{<math>x</math>-intercept} (if it exists) is the point where the function's graph crosses the <math>x</math>-axis and its <math>x</math> value is the value such that <math>f(x)=0.</math>
*The \underline{<math>y</math>-intercept} is the point where the function's graph crosses the <math>y</math>-axis and its <math>y</math> value is the value such that <math></math>y=f(0).<math></math> <math>(0,f(0))</math>
----

===Create linear functions from word problems===

Do several examples where a word problem describes a rate of change which is constant and can be represented as a linear function. Emphasize that the quantity in the problem is changing is by some {\em constant} amount over some fixed unit of time, distance, etc., and this is why it can be represented by a {\em linear} function. Discuss the meanings in practical terms of the slope and <math>y</math>-intercept. Make sure you emphasize the units of each given by the context of the example.

{{Ex-begin}}
A new Toyota RAV4 costs 21,500. The car's value depreciates linearly to 11,900 in three years. Write a formula which expresses its value <math>V</math> in dollars, in terms of its age, <math>t</math>, in years.

Input: <math>t</math> = age of the car.

Output: <math>V</math> = value of the car

Initial value (<math>b</math>): \<math>21,500.

Rate of change (</math>m$): We need to calculate. (Have students remind you of the formula).

<math>
m = \frac{21500-11900}{0-3} = \frac{9600}{-3} = -3200
</math>

So <math>V = -3200t + 21500</math>.

{{Ex-end}}

Have students do Problem 4.

Have students do Problem 5.

One thing to watch for here is that it's easy to miss including the fine as a fixed cost. Also note that the slope of the linear profit function will be 12.5 dollars per shirt, not just $12.50.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

3.5 - Angle Sum & Difference Formulas

2020-06-01T14:38:22Z

Nwakefield2: Created page with " Prior Lesson ==Objectives:== *Understand and use the half-angle, angle sum, and angle difference formulas. ---- ==Le..."

[[3.4 - Introduction to Trigonometric Identities | Prior Lesson]]

==Objectives:==

*Understand and use the half-angle, angle sum, and angle difference formulas.

----

==Lesson Guide==

===Half-Angle Formulas===

Recall that as part of Problem 3 on worksheet 12.1, students developed two alternate forms of the double-angle formula for cosine:
*$\cos(2\theta)=2\cos^2(\theta)-1$, and
*$\cos(2\theta)=1-2\sin^2(\theta)$.

By relabeling these quantities (i.e., let $\phi=2\theta$ be our "original" angle, and $\phi/2=\theta$ the "original" angle divided by $2$), we equivalently have:
*$\cos(\phi)=2\cos^2(\phi/2)-1$, and
*$\cos(\phi)=1-2\sin^2(\phi/2)$.

Rewriting these equations, we obtain:

$\cos(\phi/2)=\pm\sqrt{\frac{1+\cos(\phi)}{2}}$ and $\sin(\phi/2)=\pm\sqrt{\frac{1-\cos(\phi)}{2}}$.
These are called the \underline{half-angle formulas} for cosine and sine, respectively.

In both half-angle formulas, we have a choice: should we use the positive or negative square root? The answer will depend on the quadrant the angle lies in.

Motivate the usefulness of these formulas by telling students that we know the exact values of $\sin\theta$ and $\cos\theta$ for relatively few values of $\theta$, but with the half-angle formulas, we can find more. Give an example or two, demonstrating how we might use the half-angle formulas. For example, you could ask students to find $\sin(\pi/8)$. Make sure to emphasize how to decide whether to choose the positive or negative square root.

Have students complete Problem 1 on the worksheet.

----

===Angle Sum and Difference Formulas===

$\sin(\theta + \phi) = \sin(\theta)\cos(\phi) + \sin(\phi)\cos(\theta)$
$\sin(\theta - \phi) = \sin(\theta)\cos(\phi) - \sin(\phi)\cos(\theta)$
$\cos(\theta + \phi) = \cos(\theta)\cos(\phi) - \sin(\theta)\sin(\phi)$
$\cos(\theta - \phi) = \cos(\theta)\cos(\phi) + \sin(\theta)\sin(\phi)$

These formulas, again, help us to find the exact sine and cosine of an even larger assortment of angles. Give an example of how one might use these formulas (for example, can students find $\cos(345^{\circ})$?).

Have students complete the worksheet.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

3.4 - Introduction to Trigonometric Identities

2020-06-01T14:38:10Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Prove new trigonometric identities using..."

[[3.3 - The Laws of Sines and Cosines | Prior Lesson]] | [[3.5 - Angle Sum & Difference Formulas | Next Lesson]]

==Objectives:==

*Prove new trigonometric identities using already-proven identities and algebraic manipulation.
*Understand that there may be multiple ways of proving an identity.
*Solve trigonometric equations using identities as a tool.
;Definitions: identity

----

==Lesson Guide==

===Strategies for Proving Trigonometric Identities===

A \underline{(trigonometric) identity} is an trigonometric equation which holds true for \emph{every} possible value of any variables involved.
Note: for our purposes in this course, we will rule out any values of a variable which result in division by zero.

Give several examples of trigonometric identities, and how we might show that they hold true for every possible value of the variable. Some possibilities are given below.

----

The equation $\cot(x)=\cos(x)\csc(x)$ represents an identity, since $\cos(x)\csc(x)=\cos(x)\cdot \frac{1}{\sin(x)}=
\frac{\cos(x)}{\sin(x)}=\cot(x)$.

*In order to show the above equation represented an identity, we rewrote one side of the equation ($\cos(x)\csc(x)$) so that it matched the other side ($\cot(x)$).

----

The equation $\cos^2(x)=1-\sin^2(x)$ represents an identity, since we can add $\sin^2(x)$ to both sides in order to obtain
$\cos^2(x)+\sin^2(x)=1$, which is known to be a true statement for any value of $x$ (and in fact is the Pythagorean identity).

*Our method for establishing this identity is distinct from the first example. Here, we algebraically manipulated the desired equation to reduce it to a previously-proved identity. This is an excellent strategy for showing that an equation is an identity, and we'll use it frequently.

*What we are really saying here is that we can \emph{start} with a known identity (the Pythagorean identity), and subtract $\sin^{2}(x)$ from both sides to obtain the desired identity. Note that this subtlety will most likely be confusing to students, so whether you stress this point is up to you and where your class is.

----

Emphasize to students that there is no formula or algorithm for proving identities. It often takes experimentation using algebra and known identities.

Have students work through Problem 1 on the worksheet. This may be challenging for students, so make sure to go over the problem as a class.

----

===Double-Angle Formulas===

Introduce the idea of double-angle formulas: these will allow us to write the quantities $\cos(2\theta)$ and $\sin(2\theta)$ in terms of just $\cos(\theta)$ and $\sin(\theta)$. One option for introducing double-angle formulas is to derive $\sin(2\theta)$. If you choose to do this, a derivation is given below.

----

{{#widget:Iframe
|url=https://www.desmos.com/calculator/tiyow50lm5?embed
|width=800
|height=500
|border=0
}}

*We apply the law of sines to relate $2\theta$ and $\alpha$: $\frac{\sin(2\theta)}{2\sin(\theta)} = \frac{\sin(\alpha)}{1}$.
*We also know that $\sin(\alpha) = \cos(\theta)$ so we can conclude that $\frac{\sin(2\theta)}{2\sin(\theta)} = \cos(\theta)$, or

$\sin(2\theta) = 2\sin(\theta)\cos(\theta)$, the double-angle formula for sine.

----

Have students work through Problem 2 on the worksheet.

There is also a double-angle identity for cosine: $\cos(2\theta) = \cos^{2}(\theta)-\sin^{2}(\theta)$.

When proving identities, we often rely on previously-proven ones, such as the double-angle formulas. Encourage students to keep track of identities as they prove them, and to keep a master list somewhere in their notes for reference.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

3.3 - The Laws of Sines and Cosines

2020-06-01T14:37:43Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand and use the Law of Sines. *Understand an..."

[[3.2 - Arc Length | Prior Lesson]] | [[3.4 - Introduction to Trigonometric Identities | Next Lesson]]

==Objectives:==

*Understand and use the Law of Sines.
*Understand and use the Law of Cosines.
*Recognize potential ambiguities when using the Law of Sines.
;Definitions: Law of Sines, Law of Cosines
----

==Lesson Guide==

Up until now, we have only related trigonometric functions to right triangles. Today, the focus will shift to using our trigonometric toolset to find angles and side lengths in non-right triangles. Included in the lesson plans are derivations for the Law of Sines and the Law of Cosines. Include as many (or as few) derivations as are appropriate for your class: while they should be able to follow all derivations, at the end of the day, it is most important that they know how to use both laws.

===Standard Triangle Labeling===

Start by introducing the standard triangle labeling to students, shown below; the same letter pair gets assigned for each angle and its opposite side. It is crucial to use this convention when applying the Law of Sines or Cosines.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/8bgzfoebhg?embed
|width=500
|height=300
|border=0
}}

----

===Derivation of the Law of Sines===

Our basic process is to create right triangles from non-right triangles. Sketch a line down from angle $B$, perpendicular to side $b$, obtaining the following:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/ubp3gqwfx5?embed
|width=500
|height=300
|border=0
}}

*From the diagram, we see that $\sin(A)=h/c$, and $\sin(C)=h/a$. Solving for $h$ in both equations, we see that $c\sin(A)=h=a\sin(C)$
*Since $c\sin(A)=a\sin(C)$, we conclude $\frac{\sin(A)}{a}=\frac{\sin(C)}{c}$.
**Note that this is a slightly misleading diagram were $A$ or $C$ an obtuse angle. However, the process will still work (you'll just need to be more careful about reminding students that supplementary angles have the same sine).
*Were we to repeat the steps above, only starting with angle $A$, we would find also that $\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$.

Combining the equations from above, we obtain the Law of Sines:

$\frac{\sin(A)}{a}=\frac{\sin(B)}{b}=\frac{\sin(C)}{c}$.

Have students complete Problems 1 and 2 on their worksheets.

----

===Derivation of the Law of Cosines===

Again, sketch a line down from angle $B$, perpendicular to side $b$. Applying the Pythagorean theorem to both of the right triangles this creates, we obtain the following:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/pb1gitndre?embed
|width=500
|height=300
|border=0
}}

*$r^2+h^2=a^2$, and $(b-r)^2+h^2=c^2$.
*Solving these two equations for $h^2$, we obtain $a^2-r^2=h^2=c^2-(b-r)^2$.
*Expanding and simplifying, $a^2+b^2-2br=c^2$.
*Next, by the right-triangle definition of cosine, we also have that $\cos(C)=\frac{r}{a}$, so $a\cos(C)=r$.

Substituting, we obtain the Law of Cosines:

$a^2+b^2-2ab\cos(C) = c^2$

Starting with a different labeling, we may also show each of the following equivalent statements of the Law of Cosines:

$b^2+c^2-2bc\cos(A) = a^2$
$a^2+c^2-2ac\cos(B) = b^2$

----

Have students complete the worksheet. Problem 5 outlines the potential ambiguity with the Law of Sines. Make sure this is clear to your students once they have had a chance to work through the problem.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

----
After problem 5, you may want to discuss how we can know if we are in the ambiguous case, and how we can know if there are one or two possible triangles. Namely, the ambiguous case arises when we are given two sides and an angle $\phi$ not between them (SSA). Using Law of Sines to find another angle tells us the other angle is either $\theta$ or $\pi - \theta$. Certainly the angle $\theta$ completes a valid triangle. But if $\pi - \theta + \phi < 180$, then $\pi - \theta$ also gives a valid triangle, so there are two possible triangles.

Recitation 2

2020-06-01T14:37:09Z

Nwakefield2: /* Recitation 2 */

=Recitation 2: Functions, Exponentials and Logarithmic Functions=
==Objectives:==

==Important Notes:==

* If students try to use something other than a exponential function on problem 6 ask them if there is a better function they could use. *If they insist that linear is a good function then allow them to use a linear function but ask them where their model will break down.

==Lesson Guide==

Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (15 minutes)] Have students do Problems 1 and 2.

*[ (10 minutes)] Ask three different student to write their ''solutions'' on the board.

*[ (25 minutes)] Have students do Problems 3-4.

*[ (25 minutes)] Have students do Problems 5-6.

3.2 - Arc Length

2020-06-01T14:33:48Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand arc length as a proportion o..."

[[3.1 - Introduction to Polar Coordinates | Prior Lesson]] | [[3.3 - The Laws of Sines and Cosines | Next Lesson]]

==Objectives:==

*Understand arc length as a proportion of the circumference of a circle.
*Find the arc length given an angle on circles of varying radii.
*Find an angle given arc length and radius.
;Definitions: arc, arc length
----

==Lesson Guide==

===Arc Length Exploration===

Based on their work in Section 15.1, students are hopefully comfortable with arcs as portions of a circle.

An arc is a portion of the circumference of a circle (sketch an example here to illustrate).

Have students complete Problem 1 on the worksheet, and discuss their strategies for finding the arc lengths as a class.

===Radians and Arc Length on the Unit Circle===

An important connection is that between the ideas of arc length and radians. Have students recall that there are $2\pi$ radians in a circle and the circumference of the unit circle is $2\pi$ units.

Radians correspond to arc length on the unit circle.

Give a couple of examples (i.e. half of the way around the unit circle is $\pi$ radians, and the length of the arc corresponding to traveling half of the way around the unit circle is $\pi$ units). It may be helpful to draw pictures here to make the distinction between arc lengths and angles. Make sure that your examples don't clash with Problem 1.

-Example 1:

 
 
 
 

-Example 2:

 
 
 
 

Have students work through Problem 2.

===Arc Length on other Circles===

Suggest that we would like to find the arc length defined by an angle in radians on any circle, and give an example or two to show how we might do this.

-Example 1: Find the arc length defined by $1/2$ radians on a circle of radius $3$.

 
 
 
 

-Example 2:

 
 
 
 

Conclude that

The length $s$ of an arc defined by angle of $\theta$ radians on a circle of radius $r$ is $s = r\cdot \theta$.

Have students complete the worksheet.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

3.1 - Introduction to Polar Coordinates

2020-06-01T14:33:22Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Convert from polar coordinate..."

[[2.3 - Applications of Inverse Trigonometric Functions (Parts I & II) | Prior Lesson]] | [[3.2 - Arc Length | Next Lesson]]

==Objectives:==

*Convert from polar coordinates to Cartesian coordinates in either direction.
*Describe (simple) regions in the plane using polar coordinate inequalities.
;Definitions: polar coordinates

----

==Lesson Guide==

===What are Polar Coordinates?===
Today we will be introducing polar coordinates. Polar coordinates are really just formalizing a concept we have worked with since the beginning of class - namely, identifying points in the plane based on an angle $\theta$, and a distance $r$ from the origin (i.e., the radius of the circle they lie on).

----

Cartesian coordinates identify a point by how far left or right it lies from the origin (the $x$-coordinate), and how far above or below the origin it is (the $y$-coordinate). To contrast, polar coordinates are given as an ordered pair, $(r,\theta)$, and identify a point as being a certain distance from the origin ($r$), and corresponding to an angle from the positive $x$-axis ($\theta$, assumed to be in radians unless otherwise specified).

----

Give an example, having students help you find the Cartesian coordinates of a point given its polar coordinates, such as the one below. A picture will be helpful for students. You may want to hold off till Problem 6 to show conversion the other way.

----

The figure below shows the point $P$, whose polar coordinates are given by $(2.5,\theta)$. To find the Cartesian coordinates, we draw a right triangle as shown, and note that $\sin(pi/3)=y/2.5$ and $\cos(pi/3)=x/2.5$. Thus, the Cartesian coordinates of the point $P$ are given by $(1.25,1.25\sqrt{3})$.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/f25hikscl6?embed
|width=400
|height=400
|border=0
}}

----

Have students work through Problems 1 and 2 on the worksheet. Problem 2 emphasizes that there are multiple names for a given polar coordinate -- this will be useful to refer back to in the inverse trig sections.

=== Curves Described Using Polar Equations ===

We will not focus on this, but you can tell students that, as with Cartesian coordinates, we can plot curves using polar coordinates. While many curves can be achieved through altering trigonometric functions, we will focus here only on lines and circles, as a lead-up to the sorts of polar regions we wish to describe using inequalities (see below). Give several examples, such as:

-Example 1: The polar equation $r=1$ produces a circle of radius $1$:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/ssa0phtvie?embed
|width=300
|height=300
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-Example 2: The polar equation $\theta=\pi/8$ produces a ray through the origin (since $r$ must be nonnegative), making an angle of size $\pi/8$ with the positive $x$-axis:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/hto3xojosw?embed
|width=300
|height=300
|border=0
}}

===Describing Regions with Polar Inequalities===

We can also write polar inequalities to describe regions in the plane: sometimes, this is not easily done with Cartesian coordinates. You might show students some examples such as the following (similar to Problems 3 and 4) to illustrate this.

----
*In the shaded region below, $\theta$ is restricted to be between $\pi/2$ and $2\pi/3$ (inclusive), while within that $\theta$ range, all positive $r$ values are allowed. Thus, we can define the shaded region with the polar inequalities:

$\pi/2 \leq \theta \leq 2\pi/3$ and $0\leq r <\infty$.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/brp2e3pudo?embed
|width=500
|height=500
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*In the shaded region below, $\theta$ is unrestricted, but $r$ is (strictly) less than $4$. One possible set of polar inequalities describing the region is given below (note that we could have different $\theta$ ranges -- it's worth discussing this with your students).

$0 \leq \theta \leq 2\pi$ and $0\leq r \leq 4$.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/m1ljc3l7mo?embed
|width=500
|height=500
|border=0
}}

----

Have students work through Problems 3 and 4. Problems 5 and 6 walks students through converting from Cartesian coordinates back to polar coordinates. Make sure to emphasize that they now know how to convert in both directions.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.
----
One way to introduce the idea of polar coordinates to students is to use the analogy of "city blocks" (using Cartesian coordinates to specify walking so many city blocks left/right and up/down) vs. "walking through a cornfield" (using Polar coordinates to turn at a certain angle and walk straight for a certain distance to reach a certain point).
----
If you go over Problem 5 as a class, it is important to emphasize the relationship between the $(x,y)$ point $(\sqrt{3},1)$ which has a radius of 2 to the $(x,y)$ point $(\sqrt{3}/2, 1/2)$ on the unit circle which has a radius of 1. It might be helpful to explain that we can "scale" the point $(\sqrt{3},1)$ down by dividing the $x$ and $y$ coordinates by the radius of 2 to get the point $(\sqrt{3}/2, 1/2)$. We can then use this point to figure out what $\theta$ is by referencing the unit circle. (Or it might be more natural to go the opposite way and think about what point on the unit circle we would need to "scale" up by a factor of 2 to get to the point $(\sqrt{3}, 1)$ and then go from there. Leave a comment below if you find that one way is more helpful than another in your class.)

The reason why this is important is because we want students to recognize that they can find exact angles from points that are scaled from unit circle points. Students have the tendency to use the inverse tangent function and their calculators to find all angles and forget to think about if there might be a way to find $\theta$ exactly.
----

2.3 - Applications of Inverse Trigonometric Functions (Parts I & II)

2020-06-01T14:33:08Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Solve more complicated trigonome..."

[[2.2 - Solutions to Trigonometric Equations | Prior Lesson]] | [[3.1 - Introduction to Polar Coordinates | Next Lesson]]

==Objectives:==

*Solve more complicated trigonometric equations.
*Solve trigonometric equations on a given interval.
*Use factoring to solve equations containing multiple trigonometric functions.
----

==Lesson Guide==

There are two worksheets for this section. The second worksheet contains no new material, but is simply extra practice.

===Solving Trigonometric Equations===

We have previously seen how to solve basic trigonometric equations involving functions of just $\theta$, for example $\sin(\theta)=1/2$. Today we will consider more complex examples. Start off with an example such as the one below.

----

Solve the trigonometric equation $2\sin(3\theta-1)=-1$.

Point out that key difference between this example and our previous ones is that the input into sine is $3\theta-1$ rather than just $\theta$.

*First, we isolate the sine function: $\sin(3\theta-1)=-1/2$.
*Next, we find the solutions to this equation: since $-1/2$ corresponds to a standard angle on the unit circle, the initial solutions satisfy: $3\theta-1=7\pi/6$ and $3\theta-1=11\pi/6$.
**The quantity inside the trigonometric function, $3\theta-1$, is the input to the sine function. We set the \textit{entire input} equal to $7\pi/6$ and $11\pi/6$ to determine the initial solutions.
*Using these initial solutions, we find \textit{all possible solutions} for $3\theta-1$ by adding copies of the period of sine: $3\theta-1=7\pi/6 + 2\pi k$ and $3\theta-1=11\pi/6+2\pi k$, where $k$ is any integer. Notice we do this before solving for $\theta$.
*Solving for $\theta$ produces our final solutions of:
**$\theta=\frac{7\pi/6 +2\pi k+1}{3}=\frac{7\pi +6}{18}+\frac{2\pi}{3} k \approx 1.56 +\frac{2\pi}{3} k$, and
**$\theta=\frac{11\pi/6 +2\pi k+1}{3}=\frac{11\pi +6}{18}+\frac{2\pi}{3} k \approx 2.25 +\frac{2\pi}{3} k$.

Point out that in the example, 1.56 and 2.25 represent the initial solutions for $\theta$ (not $3\theta-1$), and the period of $2\sin(3\theta-1)$ is equal to $2\pi/3$. When teaching this process, you can instead solve for the initial solutions of 1.56 and 2.25, and then add copies of $2\pi/3$. However, the method taken above, in which copies of $2\pi$ are added \emph{before} solving for $\theta$, may be more straightforward for students.

----

Have students work through Problems 1(a) and 1(b) on the Part I worksheet.

===Solutions on a Given Interval===

We know how to find \textit{all possible} solutions to many trigonometric equations now, but sometimes we want to know the solutions to an equation \textit{only on an interval.} Illustrate this using an example such as the one below:

----

Find all solutions to $2\sin(3\theta - 1) = -1$ in the interval $[-\pi/2,3\pi / 2]$.

*We've found that all of the solutions to the equation $2\sin(3\theta - 1) = -1$ are of form $1.56 + \frac{2\pi}{3}k$ or $2.25 + \frac{2\pi}{3}k$ for some integer $k$.
*Now we test values of $k$ to figure out which solutions lie within the interval $[-\pi/2,\,3\pi/2]$:
**$k=-2$: $\theta=1.56+\frac{2\pi}{3}\cdot (-2)\approx -2.63$ and $\theta=2.25+\frac{2\pi}{3}\cdot (-2)\approx -1.94$
**$k=-1$: $\theta\approx -0.54$ and $\theta\approx 0.16$
**$k=0$: $\theta\approx 1.56$ and $\theta\approx 2.25$
**$k=1$: $\theta\approx 3.65$ and $\theta\approx 4.35$
**$k=2$: $\theta\approx 5.75$ and $\theta\approx 6.44$
*Since $[-\pi/2,\,3\pi/2]$ is approximately $[-1.57,4.71]$, we can cross off all solution sets except $k=-1$, $k=0$, and $k=1$.

We can see these solutions in the graph below:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/frveawljs7?embed
|width=800
|height=500
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}}

----

Have students work through Problems 1(c) and 2 on the Part I worksheet.

This section also includes solving equations that have more than one occurrence of sine or cosine, in which students may need to factor while solving. Give an example (for example, ask how they might solve $\sin(\theta)\tan(\theta) - \sqrt{3}\sin(\theta)=0$ for all values of $\theta$).

Have students complete the worksheet.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

2.2 - Solutions to Trigonometric Equations

2020-06-01T14:32:56Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Represe..."

[[2.1 - Generalized Sinusoidal Functions | Prior Lesson]] | [[2.3 - Applications of Inverse Trigonometric Functions (Parts I & II) | Next Lesson]]

==Objectives:==

*Represent solutions to an equation graphically.
*Understand the process of finding all $\theta$ satisfying $f(\theta)=y$, where $y$ is fixed and $f$ is a periodic function.
*Use the unit circle to solve equations of the above type for $f$ a trigonometric function.
;Definitions: initial solutions
----

==Lesson Guide==

===Solutions as Intersection Points===

Have students recall that solutions to mathematical equations can be viewed graphically as intersection points. Sketch a couple of examples, such as those shown below, with different numbers of solutions. In particular, point out (for these examples) that the quadratic equation yields two solutions, but the trigonometric function appears to have infinitely many (though not all solutions are visible).

Solutions to the equation $x^2-2=1$ are shown in the graph:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/cjxivvze9m?embed
|width=500
|height=500
|border=0
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Solutions to the equation $\cos(\theta)=2/3$ are shown in the graph:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/9pq1aau6pc?embed
|width=500
|height=500
|border=0
}}

===Solving Trigonometric Equations for All Solutions===

Have students work through Problem 1 on this worksheet. Ask students what patterns they notice in their solutions to part (c); try to guide them to see that because the graph is periodic with period 4, they may express their solutions as $1+4k$ and $3+4k$ (for any positive integer choice of $k$) if the Ferris wheel makes infinitely many rotations. Build on this idea with an example such as the following:

----

Solve the trigonometric equation $\cos(\theta)=\sqrt{3}/2$.

The key to solving this type of problem is to find the ``core" or ``initial" solutions to the equation - that is, the solutions over a single repeated segment. We can then translate them to find the remaining solutions. Consider the solutions to the equation as intersection points:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/ag23syqmkj?embed
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*$\cos(\theta) = \sqrt{3} / 2$ holds true for two angles on the unit circle: $\theta = \pi / 6$ corresponding to Solution $A$ in the diagram above and $\theta = 11\pi/6$, corresponding to Solution $B$.
**These are solutions appearing on the unit circle -- remind students to check their unit circles for exact solutions first when solving equations of this type.
*Since $\cos(\theta)$ is periodic, we can take our two initial solutions of $\theta= \pi/6$ and $\theta= 11\pi/6$, and add or subtract multiples of $2\pi$ in order to find additional solutions. For example, two additional solutions are given by: $\theta = \pi/6 - 2\pi$ and $\theta= 11\pi/6 - 2\pi$.
*To account for all solutions, we write: $\theta=\pi/6 + 2\pi k$ and $\theta=11\pi/6 + 2\pi k.$
**Emphasize that the ``$k$" here can be any integer. Show them the solutions corresponding to $k=0,-1$, and $1$ on the graph.

----

Have students work through Problem 2 on their worksheets.

Discuss the case in which the initial solutions are not angles on their unit circles. You can make up your own example here, or do Problem 3(a) as a class. Remind students that they can check their solutions, and walk them through the process of doing so in this scenario.

Make sure that students realize that it will not always be the case that we find exactly two initial solutions (there may be more or fewer) or that we always add multiples of \emph{$2\pi$} (the period of the function may be different) -- this will be explored in Problems 4 and 5 of this worksheet. Consider going over these problems as a class.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

2.1 - Generalized Sinusoidal Functions

2020-06-01T14:32:45Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand how mod..."

[[1.5 - Introduction to Inverse Trigonometric Functions | Prior Lesson]] | [[2.2 - Solutions to Trigonometric Equations | Next Lesson]]

==Objectives:==

*Understand how modifying the equation of a trigonometric function will change the period, amplitude, and midline of its graph.
*Write an equation for a trigonometric function given its graph.

----

==Lesson Guide==

Today's lesson looks at transformations of the sine and cosine functions. Students (in 103) should remember this material from Chapter 6. However, this section will serve as a reminder, specifically in the context of trigonometric functions.

The generalized sine and cosine functions have the following forms:

$A\sin(B(x - h)) + k$ and $A\cos(B(x - h)) + k$.

The most basic sine and cosine functions are when $A = 1$, $k =0$, $B =1$, and $h =0$.

----

Use Desmos to remind students how changing the values of $k$, $A$, and $B$ changes the graph of the function $A\sin(B(x-h))+k$ (set $h=0$ for now). You might have students complete a table, filling in different $k$-, $A$-, and $B$-values and finding the midline, period, and amplitude of the resulting graph. If desired, you can open the graph below and use the value sliders to give them a general idea of the effect of changing each parameter:

{{#widget:Iframe
|url=https://www.desmos.com/calculator/qwmcmrfm8i?embed
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Help students to arrive at the following conclusions (you might relate the values back to what they meant in Chapter 6 (for instance, a vertical shift changes the midline of sine and cosine):

In the function $A\sin(B(x - h)) + k$,
*The midline is $y=k$
*The amplitude is $|A|$
*The period is $2\pi/B$

Have students complete Problems 1 and 2 on the worksheet.

----

Tell students that we would also like to work backwards: given a graph of sine or cosine, we would like to be able to find a trigonometric function that gives the graph. Consider presenting an example like the one below (from Section 9.1) to motivate this process.

----

A Ferris wheel is $30$ meters in diameter, and is boarded at ground level. The wheel completes one full revolution every $4$ minutes. At time $t=0$, an individual is at the 3:00 position and is ascending. Sketch a graph of $H=f(t)$, where $H$ is the height above ground (in meters) after $t$ minutes.

{{#widget:Iframe
|url=https://www.desmos.com/calculator/nzdnnkruke?embed
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To find a trigonometric function that gives the graph above, we need to do the following:

*choose a base function (sine or cosine, in this case), determine the period, amplitude, and midline. \\

*Ask students which base function we should choose, and discuss whether to use a sine or cosine function.

*Next, have students find the period, midline, and amplitude of the Ferris wheel graph, and use these to calculate $A$, $B$, and $k$.

Remind students that after they have what they think is the correct equation, they should double-check by testing some values and making sure they actually fall on the curve.

You can also use this example to introduce horizontal shifts.

Have students complete Problem 3 in the workbook.

----

You can now either introduce students to horizontal shifts or let them explore the concept on their own by having them work through Problems 4, 5, and 6.

----

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

1.5 - Introduction to Inverse Trigonometric Functions

2020-06-01T14:32:33Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Identify in which quadrants an an..."

[[1.4 - The Tangent Function and Cofunctions | Prior Lesson]] | [[2.1 - Generalized Sinusoidal Functions | Next Lesson]]

==Objectives:==

*Identify in which quadrants an angle whose sine/cosine/tangent is given can lie.
*Use inverse trigonometric functions and a calculator to solve for an angle.
*Understand how (and why) to find multiple solutions to problems requiring the use of an inverse trigonometric function.
;Definitions: arcsine, arccosine, arctangent
----

==Lesson Guide==

===Finding Angles on the Unit Circle===

Briefly introduce students to the idea of inverse trigonometric functions and why we might want them: up until now, our computations have involved a given angle $\theta$, from which we have calculated corresponding trigonometric function values. What if instead, we know the value of a trigonometric function and want to find a corresponding angle $\theta$?
The geometry of this question is crucial to understanding how to solve it. For example, if you know $y$, how many angles $\theta$ satisfy $\sin(\theta) = y$ and in which quadrants do they lie?

Have students complete Problem 1.

Problem 1 did not involve actually solving for an angle. Introduce students to the idea of how to solve for a standard angle on the unit circle using an example or two like the following:

----

-Example 1: Find two distinct angles $\theta$ such that $\sin(\theta)=1/2$.

Note that $\sin(\theta)=1/2$ corresponds to standard angles lying on the unit circle. In this equation, we are looking for angles on the unit circle whose corresponding $y$-value is $1/2$. We use the unit circle to find: $\theta=\pi/6$ or $\theta=5\pi/6$.

----

Have students complete Problem 2.

===Non-Standard Angles on the Unit Circle===

Not all angles are standard angles on the unit circle. Use an example to introduce the process of finding $\theta$ when it is a non-standard angle on the unit circle. A sample example is given below:

----

- Example 2: Find two distinct angles $\theta$ such that $\sin(\theta)=2/3$.

Because $\sin(\theta)$ is positive, $\theta$ may lie in either quadrant I or II. However, no standard angle on the unit circle produces $\sin(\theta)=2/3$.
We first need an additional tool - namely, inverse trigonometric functions.

$\arcsin(2/3)=\sin^{-1}(2/3)=\theta$ implies that $\sin(\theta)=2/3$ (note that there are two forms of stating the inverse trig function).
*Note that $\sin^{-1}(\theta)$ is not equal to $\frac{1}{\sin(\theta)}$.

Using a calculator, we compute that $\sin^{-1}(2/3)\approx 0.73=\theta$. (Make sure everyone is in radian mode on their calculators!)

However, as with the first example of $\sin(\theta)=1/2$, we need to find two solutions on the interval $[0,2\pi]$. Unfortunately, the calculator only gives us one solution. How can we find another? Sketch the unit circle and show where the other angle should lie, as shown in the figure below. This agrees with our statement that $\theta$ may lie in either quadrant I or II. We conclude that $\theta=0.73$ and $\theta=\pi-0.73\approx 2.41$ are our two distinct angles within $[0,2\pi]$ satisfying $\sin(\theta)=2/3$. Remind students that they can check their work by evaluating $\sin(0.73)$ and $\sin(2.41)$, and making sure the solutions are close to $2/3$ (remind them that rounding will make the solutions approximate).

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|url=https://www.desmos.com/calculator/nm5dw0zv46?embed
|width=500
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}}

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Have students complete Problems 3-5. Discuss Problem 5 as a class.

Have students complete the worksheet.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

1.4 - The Tangent Function and Cofunctions

2020-06-01T14:32:19Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand the relationship..."

[[1.3 - The Sine & Cosine Functions | Prior Lesson]] | [[1.5 - Introduction to Inverse Trigonometric Functions | Next Lesson]]

==Objectives:==

*Understand the relationship between the tangent function and sine and cosine.
*Understand the relationship between tangent and right triangles.
*Be able to use the definitions of reciprocal trigonometric functions.
*Determine if two functions are cofunctions.
;Definitions: tangent, secant, cosecant, cotangent, cofunction
----

==Lesson Guide==

===The Tangent Function===
Start by introducing the tangent function. There are, again, two different definitions. You can introduce them at the same time, or put some examples in between the two definitions.

Given an angle $\theta$ (in either degrees or radians), we define $\tan(\theta)=\frac{\sin(\theta)}{\cos(\theta)}$.

Or, we may define tangent as follows:

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$\tan(\theta)=\frac{opposite}{adjacent}=\frac{o}{a}$. This is the "TOA" part of the mnemonic SOH-CAH-TOA, and should be read as "tangent is opposite over adjacent." Again, these two definitions agree, since $\frac{\sin(\theta)}{\cos(\theta)}=\frac{o/h}{a/h}=\frac{o}{a}$.

===The Graph of Tangent===

Have students calculate the value of $\tan(\theta)$ for each of the angles on their units circles. As a class, use these values to sketch a graph of the tangent function (again, inputs are in radians). Point out to students that the graph of $\tan(\theta)$, despite being periodic, does not have a midline or amplitude. This is because it is not a wave-like periodic function.

Have students work through Problems 1-2 on their worksheets.

===Reciprocal Trig Functions & Cofunctions===

There are three other commonly used trigonometric functions, defined below:
Secant: $\sec(\theta) = \frac{1}{\cos(\theta)}$
Cosecant: $\csc(\theta) = \frac{1}{\sin(\theta)}$
Cotangent: $\cot(\theta) = \frac{1}{\tan(\theta)}$

*Two functions are called \underline{cofunctions} if they are equal on complementary angles (i.e., angles adding to $90^\circ$, or equivalently $\pi/2$ radians).
*Sine and cosine are examples of cofunctions (hence the ``co'' in ``cosine'').

----

You may illustrate this relationship as follows: note that if two angles are complementary, they may be the acute angles of a right triangle. For instance, in the diagram below, $\phi+\theta=90^{\circ}$. Using our SOH-CAH definitions, $\sin(\theta)=a/c$, and $\cos(\phi)=a/c$.

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|url=https://www.desmos.com/calculator/iagy0mmwfl?embed
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Have students complete Problems 3 and 4 on their worksheets. You may want to remind students what even and odd functions are before doing Problem 4.

Just as sine has cofunction cosine, secant and tangent also have cofunctions:
$\sin(\theta)=\cos\left(\frac{\pi}{2}-\theta\right)$,
$\sec(\theta)= \csc\left(\frac{\pi}{2}-\theta\right)$, and
$\tan(\theta)=\cot\left(\frac{\pi}{2}-\theta\right).$

Have students complete the remainder of the worksheet.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

1.3 - The Sine & Cosine Functions

2020-06-01T14:31:56Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Understand the purpose of the sine..."

[[1.2 - Introduction to the Unit Circle | Prior Lesson]] | [[1.4 - The Tangent Function and Cofunctions | Next Lesson]]

==Objectives:==

*Understand the purpose of the sine and cosine functions.
*Understand the relationship between sine and cosine and right triangles.
*Use the unit circle to find the sine and cosine of an angle.
*Use the Pythagorean identity.
;Definitions: sine, cosine
----

==Lesson Guide==

===The Unit Circle===
Given an angle $\theta$ (in either degrees or radians), and the $(x,y)$-coordinates of the corresponding point
on the unit circle, we define $\cos(\theta)=x$, and $\sin(\theta)=y$.
*Note that sine and cosine are functions which take angles as inputs.

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Give several examples from the unit circle, which students should have completely filled out, to emphasize the relationship between sine and cosine and coordinates on the unit circle.

===SOH-CAH-TOA===

Students may have also seen sine and cosine defined in the context of right triangles, which you should also introduce:

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Where $\sin(\theta)=\frac{opposite}{hypotenuse}=o/h$, and $\cos(\theta)=\frac{adjacent}{hypotenuse}=a/h$.

This gives rise to the popular mnemonic SOH-CAH-TOA, where the first two abbreviations stand for "sine is opposite over hypotenuse," and "cosine is adjacent over hypotenuse." (TOA relates to another function which we'll look at later.)

----

Have students find the sine and cosine of an angle in an example right triangle, such as the one below:

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|url=https://www.desmos.com/calculator/4qzkfe6e8n?embed
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Emphasize to students that the two definitions presented are the same! Point out that on the unit circle, the length of the hypotenuse is equal to 1, so the definitions match.

Have students work through Problems 1 and 2 on the worksheet. Students should first try to find the sine and cosine of an angle by using their unit circles. Only if the angles aren't on the unit circle should they use their calculators. Make sure students know how to switch back and forth between degree and radian mode on their calculators.

===Graphs of Sine and Cosine===
Introduce the graphs of the sine and cosine functions (make sure to emphasize that inputs are in radians). You may choose to plot a few values from the unit circle first, and then connect them. This is a good chance to emphasize the relationship between sine and cosine and coordinates on the unit circle.

Have students work through Problem 5 on their worksheet, then discuss their observations as a whole class.

Have students complete the worksheet. Note that Problem 6 deals with the Pythagorean Identity so you should be sure to go over this concept with your class. Part 6a may take students quite some time. Best to have particular tables make certain calculation and have them report back to the class.

Problems 7 and 8 deal with finding the coordinates of a point on a circle of radius larger than 1 and circles not centered at the origin.

At some point during lecture, you should tell students that to represent a trigonometric function raised to a power, for example
$(\sin(x))^2$, we write $\sin^2(x)$, rather than $\sin(x)^2$. A good place for this is before Problem 6.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

- Some students seemed to get tripped up with relating the (x,y) = (r cos(theta), r sin(theta)) to right triangles when the triangles were not in the first quadrant. They thought "Shouldn't I be calculating sin and cos using the angle be *inside* the triangle (instead of the angle from the x-axis)?" So maybe doing an example like this would be helpful.

1.2 - Introduction to the Unit Circle

2020-06-01T14:31:36Z

Nwakefield2: Created page with " Prior Lesson | Next Lesson ==Objectives:== *Convert between radians and degrees. *Fill in the unit c..."

[[1.1 - Periodic Functions | Prior Lesson]] | [[1.3 - The Sine & Cosine Functions | Next Lesson]]

==Objectives:==

*Convert between radians and degrees.
*Fill in the unit circle.
;Definitions: the unit circle, radians
----

==Lesson Guide==

===Introduction to Radians===

Start by introducing students to the idea of the unit circle. They have a blank copy of the unit circle in their worksheet (Problem 6). Let them know that the goal for the class will be to fill in the details of the unit circle: the angles as well as the coordinates of the points.

The $\underline{unit circle}$ is a circle of radius 1, centered at the origin. When measuring an angle around the unit circle,
we travel in the counter-clockwise direction, starting from the positive $x$-axis. A negative angle is measured in the opposite,
or clockwise, direction. A complete trip around the unit circle amounts to a total of 360 degrees.

Have students fill in the degree measures of the angles marked on the unit circle, given the following information: 12 of the angles are obtained by moving $360/12=30^{\circ}$ at a time around the unit circle, and the other 4 are obtained by bisecting each of the quadrants (you may have to remind students that each quadrant constitutes $90^{\circ}$.

Introduce students to the idea of using radians as an alternative angle measure. One option is to tell students that radians are related to the circumference of the unit circle. Ask them what the circumference of the unit circle is, and then tell them that a complete trip around the unit circle is $2\pi$ radians:

Radians arise from looking at angles as a fraction of the circumference of the unit circle; a complete trip around the unit circle
amounts to a total of $2\pi$ radians.
*Even though radians are related to the circumference of the {\bf unit} circle, radians are a unit for measuring angles.
Make sure students are aware of this distinction.

Have students fill in the radian measures of each of the angles on their unit circles. You may want to give a couple of examples converting degrees to radians before having them do this, or you may let them explore this relationship on their own.

Solidify the relationship between degrees and radians. You can use the unit circle as a jumping-off point, and then do an example or two converting back and forth with angles that are not on their unit circles. Tell students that when working with physical quantities (angle above a horizon, interior angle of a triangle, etc.), we tend to work with degrees. On the other hand, when working in abstract settings (graphing functions, solving equations, etc.), we tend to work with radians. Our default angle measurements will be made in radians. Unless degrees are specified for a particular problem, students should assume that all angle measures are in radians.

Have students complete Problems 1-5 on the worksheet. Regroup to make sure that everyone has arrived at the correct conclusions in Problem 3.

===Filling in the Unit Circle===

Work on filling in the coordinates of the points in the the first quadrant of the unit circle. How much detail you go into with the derivations is up to you, but students will need to have at least the first quadrant of their unit circles completed by the end of class. Below are some examples of how you might present the derivations:

----

Let's identify the $(x,y)$-coordinates of the point on the unit circle corresponding to an angle of $45^\circ$ or $\pi/4$ radians.
Consider the following illustration:

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|height=600
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*Since we are working on the unit circle, our radius is 1 unit, so $r = 1$.
*Recall that the Pythagorean theorem states that, for right triangles \textit{only}, $a^2+b^2=c^2,$ where $a$ and $b$ represent the lengths of the triangle's legs, and $c$ represents the length of the hypotenuse. How can we rewrite this using the variables from our illustration?
*Since $\theta = \pi/4$, $y = x$. We can use this, along with $r=1$, to solve for $x$: $x=\frac{\sqrt{2}}{2}$.
*We conclude that $y = \frac{\sqrt{2}}{2}$ as well.
*Have students fill in these values on their unit circles.

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Now let's identify the $(x,y)$ point which corresponds to an angle of $\pi/6$ radians or 30 degrees on the unit circle. Consider the following illustration:

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|url=https://www.desmos.com/calculator/ecpfolno8g?embed
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*The \emph{large} triangle shown is an equilateral triangle since $r$ and $s$ are equal (why?) and one of the angles is $60^\circ$ (which angle?). So we know that the $y$-coordinate of the point $P$ is $y=1/2$.
*Now we can use the Pythagorean theorem again to find $x$: $r^2 = x^2 + y^2$, so $x = \frac{\sqrt{3}}{2}$.
*Have students fill in these values on their unit circles.
*From here, you could have students follow a similar process to derive the coordinates for the angle $\pi/3$.

----

Have students use what they know about the coordinates in the first quadrant to fill out the remainder of the unit circle. Walk among groups to ensure that everyone arrives at a consensus. Encourage students to commit the unit circle to memory.

==Comments==
----
In this area, you should feel free to add any comments you may have on how this lesson has gone or what other instructors should be aware of.

----

One way to explain the relationship between degrees and radians is by comparing them to using feet vs. meters when measuring length or gallons vs. liters when measuring the volume of a liquid. Both degrees and radians measure the same thing - angles, but are different units of measurement.

----

There are two possible answers to Problem 5 in the workbook. You might want to observe if any of your students notice this while they are working on this problem.

----

Recitation 15

2020-06-01T14:30:07Z

Nwakefield2: Created page with " =The First and Second Derivative= ==Objectives:== * Students will be able to find and classify critical points. ==Important Notes:== ==Lesson Plan== *[ (5 minutes)]..."

=The First and Second Derivative=
==Objectives:==

* Students will be able to find and classify critical points.

==Important Notes:==

==Lesson Plan==
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students work through problems 1-3. The point of these problems is not algebra but calculus, the problems should have easy algebra. If a student gets stuck on algebra give them the assistance they need to focus on the calculus.

*[ (5 minutes)] Have students solve number 4. This is supposed to be a graphical reasoning problem not an algebraic problem. The point is not to find all the extrema, but so say something like, $-\cos(x)$ crosses $2e^x$ an infinite number of times when $x<0$.

*[ (10 minutes)] Have students work together on problem 5.

*[ (10 minutes)] Have students work together on problem 6. This problem requires algebra.

*[ (15 minutes)] Have students work together on problem 7. Make sure to look at problem 7 before you go into the classroom, it can be tricky.

*[ (20 minutes)] Assign different people the task of presenting their answers to each problem to the entire recitation.

Recitation 16

2020-06-01T14:29:46Z

Nwakefield2: Created page with " =Global Extrema= ==Objectives:== * Students will be able to find and classify critical points and decide if they are local or global maxima. ==Important Notes: == ==Le..."

=Global Extrema=
==Objectives:==

* Students will be able to find and classify critical points and decide if they are local or global maxima.

==Important Notes: ==

==Lesson Notes==
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (20 minutes)] Have students work through problems 1-2. These problems are mostly just computation.

*[ (10 minutes)] Have students solve number 3 the purpose of this problem is to show students that all the information they are using can be very useful for drawing more accurate graphs.

*[ (10 minutes)] Have students solve number 4. Again, the purpose of this problem is to show students that all the information they are using can be very useful for drawing more accurate graphs.

*[ (15 minutes)] Have students finish the worksheet.

*[ (15 minutes)] Assign different people the task of presenting their answers to each problem to the entire recitation.

Recitation 17

2020-06-01T14:29:27Z

Nwakefield2: Created page with " =Global Extrema Part 2= ==Objectives:== * Students will be able to find and classify critical points and decide if they are local or global maxima. ==Important Notes:==..."

=Global Extrema Part 2=
==Objectives:==

* Students will be able to find and classify critical points and decide if they are local or global maxima.

==Important Notes:==

==Lesson Plan==
*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (10 minutes)] Have students work through problem1 . Make sure you have tried the problem before you go into the classroom as this one can be somewhat challenging.

*[ (10 minutes)] Have students solve number 2 the purpose of this problem is to show students that all the information they are using can be very useful for graphin.

*[ (10 minutes)] If a group of students is really struggling have them complete problem 4 and skip problem 4, if everything is going fine then have the group do problem 4.

*[ (10 minutes)] Have students work through problem 7 . 7C is especially hard and the hint sometimes has had an error. The hint should read something like \[-\frac{270}{x^2}+\frac{20 \log (x)}{x^2}+\frac{1}{10}=0\] when $x \approx 44.0773$. This is a hard problem, but something that they should at least be able to set up.

*[ (20 minutes)] Every groups needs to have a solution to problem 8 and 9. These problems are exam level problems and they should expect to see something like this on an exam.

*[ (10 minutes)] Go over problems 8 and 9 slowly and in detail.

Recitation 18

2020-06-01T14:29:05Z

Nwakefield2: Created page with " =Recitation 18= ==Objectives:== * Students will set-up and solve related rates problems. ==Important Notes: == ==Lesson Plan== *[ (5 minutes)] Welcome the class to re..."

=Recitation 18=
==Objectives:==

* Students will set-up and solve related rates problems.

==Important Notes: ==
==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (60 minutes)] Have students work through the packet, these are challenging problems and require them to set-up models. There is not really anything to say here other then help the students where they need help but do not give away too much. Tell students that setting up the problem is the skill they need to work on, the calculus is easy. Just a warning, problem 6 should really be the speed of the tip of the shadow with reference to the pole, it is easy to make the mistake of measuring the speed relative to the person.

As a note, problem one may seem like it has nothing to do with related rates. However, implicit differentiation is the backbone of related rates problems. The purpose of problem 1 is to re-familiarize students with implicit differentiation.

*[ (10 minutes)] Go over two problems that you think are meaningful.

Recitation 19

2020-06-01T14:28:48Z

Nwakefield2: Created page with " =Recitation 19= ==Objectives:== * Students will be prepared for the second exam ==Important Notes: == ==Lesson Plan== * Make sure to look at the exam review section..."

=Recitation 19=
==Objectives:==

* Students will be prepared for the second exam

==Important Notes: ==

==Lesson Plan==

* Make sure to look at the exam review section in the back of the course packet to see if there are any problems that are irrelevant given the pace of the course. Also check to make sure there are not problems blatantly missing based on the pace of the course.

*[ (75 minutes)] I find it best to tell students that for a review day if 80\% or more of the class comes with the review material completely done then I will go over the answers. This tends to help the students study with a purpose and then learn a lot from your act of going over everything in detail. I strongly recommend that you write the solution to each problem just as you would expect the solution to be written on the exam. This also helps students to see what a good solution will look like.

Recitation 20

2020-06-01T14:28:30Z

Nwakefield2: Created page with " =Recitation 20= ==Objectives:== * Students should get some practice recognizing an applying L'Hopitals rule. ==Important Notes: == ==Lesson Plan== *[ (5 minutes)] Welcome..."

=Recitation 20=
==Objectives:==
* Students should get some practice recognizing an applying L'Hopitals rule.

==Important Notes: ==

==Lesson Plan==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (20 minutes)] Have students work through problem 1

*[ (15 minutes)] Have students work through problem 2. This is something that they have probably not seen before but is something from their homework.

*[ (15 minutes)] Have students work through problem 3. If you have never done a problem like this then you should make sure that you also know how to make sense of it by calculating slopes.

*[ (15 minutes)] Have students work through problems 4, 5, and 6.

*[ (10 minutes)] Go over two problems that you think are meaningful.

Recitation 21

2020-06-01T14:28:10Z

Nwakefield2: Created page with "=Recitation 21= ==Objectives:== * Students will be able to recognize parametric curves. * Students will be able to calculate rates of change involving parametric curves. ==..."

=Recitation 21=
==Objectives:==
* Students will be able to recognize parametric curves.
* Students will be able to calculate rates of change involving parametric curves.

==Important Notes:==

*[ (5 minutes)] Welcome the class to recitation. Ask students if they have filled in the notes section of the packet. Tell students that they should be filling in the notes section before coming to recitation.

*[ (30 minutes)] Have students work through problems 1-3.

*[ (10 minutes)] Have students work through problem 4..

*[ (10 minutes)] Have students work through problem 5. Students really struggle to think about the same problem in multiple ways so you should expect them to really struggle with part b.

*[ (5 minutes)] Have students work through problem 6..

*[ (10 minutes)] Have students work through problem 7. Make sure students know how to do this on their graphing calculator and also by plotting points.

*[ (10 minutes)] Have students work through problem 8. Many students are likely to feel completely lost. With a student who is lost you should take a step back and have them find the tangent line to something like $y=x^2$ and then ask them what information they would need to do something similar here.

* Strong students can continue working onto problems 9 and 10.

*[ (15 minutes)] Go over problem 8 through an interactive lecture.