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

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[[ 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>.

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==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.
<br>

{| 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.

3.1: Exponential Functions - Revision history

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