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2020-06-01T14:44:43Z

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

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[[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:

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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?
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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?

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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. ->

5.2: Vertical & Horizontal Shifts - Revision history

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