3.3: Graphs of Exponential Functions - Revision history

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

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[[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.