3.2: Comparing Exponential & Linear Growth
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.
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This is a sentence that is in place to generate space in the table |
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 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(0)=750} , 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(2)=1000} . 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 750=P_0(b)^0} which means that 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_0=750} since 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=1} . From here we can say 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 1000 = 750 (b)^2 } 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 \frac{4}{3} = b^2 } 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 \frac{2}{\sqrt{3}} =b } Therefore, we can model the situation using the equation 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)=750\left(\frac{2}{\sqrt{3}} \right)^t} |
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.
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This is a sentence that is in place to generate space in the table |
Which is linear and which is exponential? If linear, what is the constant rate of change? If exponential, what is the growth factor?
Discuss/write on board: *Exponential with a constant growth factor of 2; *Linear with a constant rate of change 10. |
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
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This is a sentence that is in place to generate space in the table |
On the board, write a completed table such as the following 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 f(t)}
is linear, 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 g(t)}
is exponential, 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 h(t)}
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.
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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
