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Precalculus for Team-Based Inquiry Learning PREVIEW Edition

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Section 5.2 Graphs of Exponential Functions (EL2)

Subsection 5.2.1 Activities

Activity 5.2.1.

Consider the function \(f(x)=2^x\text{.}\)
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(a)
Fill in the table of values for \(f(x)\text{.}\) Then plot the points on a graph.
\(x\) \(f(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
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Answer.
\(x\) \(f(x)\)
\(-2\) \(\dfrac{1}{4}\)
\(-1\) \(\dfrac{1}{2}\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
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(b)
What seems to be happening with the graph as \(x\) goes toward infinity? Plug in large positive values of \(x\) to test your guess, then describe the end behavior.
  1. As \(x \to \infty\text{,}\) \(f(x) \to -\infty\text{.}\)
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  2. As \(x \to \infty\text{,}\) \(f(x) \to -2\text{.}\)
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  3. As \(x \to \infty\text{,}\) \(f(x) \to 0\text{.}\)
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  4. As \(x \to \infty\text{,}\) \(f(x) \to 2\text{.}\)
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  5. As \(x \to \infty\text{,}\) \(f(x) \to \infty\text{.}\)
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Answer.
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(c)
What seems to be happening with the graph as \(x\) goes toward negative infinity? Plug in large negative values of \(x\) to test your guess, then describe the end behavior.
  1. As \(x \to-\infty\text{,}\) \(f(x) \to -\infty\text{.}\)
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  2. As \(x \to -\infty\text{,}\) \(f(x) \to -2\text{.}\)
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  3. As \(x \to -\infty\text{,}\) \(f(x) \to 0\text{.}\)
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  4. As \(x \to -\infty\text{,}\) \(f(x) \to 2\text{.}\)
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  5. As \(x \to -\infty\text{,}\) \(f(x) \to \infty\text{.}\)
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Answer.
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(d)
Complete the graph you started in Task 5.2.1.a, connecting the points and including the end behavior you’ve just determined.
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Answer.
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(e)
Does your graph seem to have any asymptotes?
  1. No. There are no asymptotes.
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  2. There is a vertical asymptote but no horizontal one.
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  3. There is a horizontal asymptote but no vertical one.
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  4. The graph has both a horizontal and vertical asymptote.
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Answer.
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(g)
Find the domain and range of \(f(x)\text{.}\) Write your answers using interval notation.
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Answer.
Domain: \((-\infty,\infty)\text{,}\) Range: \((0, \infty)\)
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(h)
Find the interval(s) where \(f(x)\) is increasing and the interval(s) where \(f(x)\) is decreasing. Write your answers using interval notation.
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Answer.
Increasing: \((-\infty,\infty)\text{,}\) Decreasing: nowhere
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Remark 5.2.2.

The graph of an exponential function \(f(x)=b^x\) where \(b>1\) has the following characteristics:
  • Its domain is \((-\infty,\infty)\) and its range is \((0,\infty)\text{.}\)
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  • It is an exponential growth function; that is it is increasing on \((-\infty,\infty)\text{.}\)
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  • There is a horizontal asymptote at \(y=0\text{.}\) There is no vertical asymptote.
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  • There is a \(y\)-intercept at \((0,1)\text{.}\) There is no \(x\)-intercept.
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Activity 5.2.3.

Consider the function \(g(x)=\left(\dfrac{1}{2}\right)^x\text{.}\)
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(a)
Fill in the table of values for \(g(x)\text{.}\) Then plot the points on a graph.
\(x\) \(g(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
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Answer.
\(x\) \(g(x)\)
\(-2\) \(4\)
\(-1\) \(2\)
\(0\) \(1\)
\(1\) \(\dfrac{1}{2}\)
\(2\) \(\dfrac{1}{4}\)
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(b)
What seems to be happening with the graph as \(x\) goes toward infinity? Plug in large positive values of \(x\) to test your guess, then describe the end behavior.
  1. As \(x \to \infty\text{,}\) \(g(x) \to -\infty\text{.}\)
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  2. As \(x \to \infty\text{,}\) \(g(x) \to -2\text{.}\)
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  3. As \(x \to \infty\text{,}\) \(g(x) \to 0\text{.}\)
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  4. As \(x \to \infty\text{,}\) \(g(x) \to 2\text{.}\)
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  5. As \(x \to \infty\text{,}\) \(g(x) \to \infty\text{.}\)
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Answer.
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(c)
What seems to be happening with the graph as \(x\) goes toward negative infinity? Plug in large negative values of \(x\) to test your guess, then describe the end behavior.
  1. As \(x \to-\infty\text{,}\) \(g(x) \to -\infty\text{.}\)
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  2. As \(x \to -\infty\text{,}\) \(g(x) \to -2\text{.}\)
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  3. As \(x \to -\infty\text{,}\) \(g(x) \to 0\text{.}\)
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  4. As \(x \to -\infty\text{,}\) \(g(x) \to 2\text{.}\)
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  5. As \(x \to -\infty\text{,}\) \(g(x) \to \infty\text{.}\)
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Answer.
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(d)
Complete the graph you started in Task 5.2.3.a, connecting the points and including the end behavior you’ve just determined.
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Answer.
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(e)
What are the equations of the asymptote(s) of the graph?
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Answer.
\(y=0\)
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(f)
Find the domain and range of \(f(x)\text{.}\) Write your answers using interval notation.
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Answer.
Domain: \((-\infty,\infty)\text{,}\) Range: \((0, \infty)\)
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(g)
Find the interval(s) where \(f(x)\) is increasing and the interval(s) where \(f(x)\) is decreasing. Write your answers using interval notation.
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Answer.
Increasing: nowhere, Decreasing: \((-\infty,\infty)\)
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Activity 5.2.4.

Consider the two exponential functions we’ve just graphed: \(f(x)=2^x\) and \(g(x)=\left(\dfrac{1}{2}\right)^x\text{.}\)
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(a)
How are the graphs of \(f(x)\) and \(g(x)\) similar?
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Answer.
Answers could include basic shape, asymptote, domain, range.
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(b)
How are the graphs of \(f(x)\) and \(g(x)\) different?
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Answer.
Answers could include reflection over \(y\)-axis, one is increasing, one is decreasing.
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Remark 5.2.5.

We can now update Remark 5.2.2 so that it includes all values of the base of an exponential function.
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The graph of an exponential function \(f(x)=b^x\) has the following characteristics:
  • Its domain is \((-\infty,\infty)\) and its range is \((0,\infty)\text{.}\)
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  • Aside
    If \(b>1\text{,}\) \(f(x)\) is increasing on \((-\infty,\infty)\) and is an exponential growth function. If \(0 < b < 1\text{,}\) \(f(x)\) is decreasing on \((-\infty,\infty)\) and is an exponential decay function.
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  • There is a horizontal asymptote at \(y=0\text{.}\) There is no vertical asymptote.
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  • There is a \(y\)-intercept at \((0,1)\text{.}\) There is no \(x\)-intercept.
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Activity 5.2.6.

Let’s look at a third exponential function, \(h(x)=2^{-x}\text{.}\)
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(a)
Before plotting any points or graphing, what do you think the graph might look like? What sort of characteristics might it have?
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Answer.
With the \(2\) as the base, students may assume exponential growth. But the negative in the exponent may change their mind if they remember transformations!
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(b)
Fill in the table of values for \(h(x)\text{.}\) Then plot the points on a graph.
\(x\) \(h(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)
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Answer.
\(x\) \(h(x)\)
\(-2\) \(4\)
\(-1\) \(2\)
\(0\) \(1\)
\(1\) \(\dfrac{1}{2}\)
\(2\) \(\dfrac{1}{4}\)
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(c)
This function \(h(x)\) looks to be the same as a function we looked at previously. Use properties of exponents to rewrite \(h(x)\) in a different way.
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Answer.
\(2^{-x} = (2^{-1})^x = \left(\dfrac{1}{2} \right)^x \)
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(d)
In addition to plotting points, we can use transformations to graph. If we consider \(f(x)=2^x\) to be the parent function, what transformation is needed to graph \(h(x)=2^{-x}\text{?}\)
  1. A vertical stretch.
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  2. A horizontal stretch.
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  3. A reflection over the \(x\)-axis.
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  4. A reflection over the \(y\)-axis.
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Answer.
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Activity 5.2.8.

Let \(f(x)=4^{x}\text{.}\)
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(a)
Graph \(f(x)\text{.}\)
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Answer.
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(b)
Match the following functions to their graphs.
  • \(\displaystyle g(x)= -4^x \)
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  • \(\displaystyle h(x)= 4^{-x} \)
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  • \(\displaystyle j(x)= 4^{x+1} \)
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  • \(\displaystyle k(x)= 4^x +1 \)
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Answer.
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(c)
Find the domain, range, and equation of the asymptote for the parent function \(\left(f(x)\right)\) and each of the four transformations \(\left(g(x), h(x), j(x), \text{ and } k(x)\right)\text{.}\)
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Answer.
\(f(x)\text{:}\)
\(g(x)\text{:}\)
\(h(x)\text{:}\)
\(j(x)\text{:}\)
\(k(x)\text{:}\)
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(d)
Which of the transformations affected the domain of the exponential function? Select all that apply.
  1. A vertical shift.
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  2. A horizontal shift.
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  3. A reflection over the \(x\)-axis.
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  4. A reflection over the \(y\)-axis.
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  5. None of these.
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Answer.
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(e)
Which of the transformations affected the range of the exponential function? Select all that apply.
  1. A vertical shift.
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  2. A horizontal shift.
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  3. A reflection over the \(x\)-axis.
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  4. A reflection over the \(y\)-axis.
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  5. None of these.
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Answer.
A and C
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(f)
Which of the transformations affected the asymptote of the exponential function? Select all that apply.
  1. A vertical shift.
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  2. A horizontal shift.
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  3. A reflection over the \(x\)-axis.
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  4. A reflection over the \(y\)-axis.
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  5. None of these.
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Answer.
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Activity 5.2.9.

Consider the function \(f(x)=e^{x}\text{.}\)
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(a)
Graph \(f(x)=e^{x}\text{.}\) First find \(f(0)\) and \(f(1)\text{.}\) Then use what you know about the characteristics of exponential graphs to sketch the rest. Then state the domain, range, and equation of the asymptote. (Recall that \(e \approx 2.72\) to help estimate where to put your points.)
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Answer.
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(b)
Sketch the graph of \(g(x)=e^{x-2}\) using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.
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Answer.
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(c)
Sketch the graph of \(h(x)=-3e^x\) using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.
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Answer.
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  • Transformations: vertical stretch of 3, reflection over \(x\)-axis
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  • Domain: \((-\infty,\infty)\)
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  • Range: \((-\infty,0)\)
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  • Asymptote: \(y=0\)
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(d)
Sketch the graph of \(g(x)=e^{-x}-4\) using transformations. State the transformation(s) used, the domain, the range, and the equation of the asymptote.
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Answer.
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  • Transformations: reflection over \(y\)-axis, shift down 4
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  • Domain: \((-\infty,\infty)\)
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  • Range: \((-4,\infty)\)
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  • Asymptote: \(y=-4\)
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Activity 5.2.10.

Graph each of the following exponential functions. Include any asymptotes with a dotted line. State the domain, the range, the equation of the asymptote, and whether it is growth or decay.
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(a)
\(f(x)=3^x\)
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Answer.
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(b)
\(f(x)=6^{-x}\)
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Answer.
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(c)
\(f(x)=\dfrac{1}{5}^{x-2}\)
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Answer.
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T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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(d)
\(f(x)=\dfrac{1}{3}^{x}+4\)
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Answer.
Diagram Exploration Keyboard Controls
Key Action
Enter, A Activate keyboard driven exploration
B Activate menu driven exploration
Escape Leave exploration mode
Cursor down Explore next lower level
Cursor up Explore next upper level
Cursor right Explore next element on level
Cursor left Explore previous element on level
X Toggle expert mode
W Extra details if available
Space Repeat speech
M Activate step magnification
Comma Activate direct magnification
N Deactivate magnification
Z Toggle subtitles
C Cycle contrast settings
T Monochrome colours
L Toggle language (if available)
K Kill current sound
Y Stop sound output
O Start and stop sonification
P Repeat sonification output
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Subsection 5.2.2 Exercises

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