3.1: Exponential Functions
Contents
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t} .
Important Items
Definitions:
exponential function, growth factor, growth rate
Notes:
While the old textbook used the notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q=a(b)^t} , we use function notation Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(t)=a(b)^t} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(b)^t} whenever possible. We want to emphasize functional notation. Furthermore, try to always use parentheses to denote multiplication, i.e., Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(b)^t} , rather than Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a\cdot b^t} or Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=ab^t} . This makes it easier to transition to talking about the growth factor Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(b)^t} and the annual growth rate Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} in Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(t)=a(1+r)^t} .
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.
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Instructor Example |
Suggested Example | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
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This is a sentence that is in place to generate space in the table |
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t}
years after the money was originally deposited.
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. |
Definition: An exponential function is a function of the form Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(t)=a(b)^t,} (Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a\not=0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>0} , Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b\neq1} ), where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a} is the initial value of Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle Q(t)} (at Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t=0} ) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} , the base, is the growth factor. The growth rate is Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b-1} .
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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle a>0} .
*If there is exponential growth, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b>1}
.
*If there is exponential decay, then Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 0<b<1}
.
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.
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Instructor Example |
Suggested Example |
|---|---|
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This is a sentence that is in place to generate space in the table |
The grades of six students are given in problem 4. Let's look at the graphs of the equations given by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(t)=97(1.001)^t} and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle P(t)=85(0.89)^t} .
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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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b} and growth rate in your example, and be sure to write it down formally:
Growth Factor: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b}
Growth Rate: Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b-1}
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.
