Graphs of Exponential Functions (EL2)

Section 5.2 Graphs of Exponential Functions (EL2)

Objectives

Graph exponential functions and determine the domain, range, and asymptotes.

Subsection 5.2.1 Activities

Activity 5.2.1.

Consider the function \(f(x)=2^x\text{.}\)

(a)

Fill in the table of values for \(f(x)\text{.}\) Then plot the points on a graph.

\(x\)	\(f(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)

Answer.

\(x\)	\(f(x)\)
\(-2\)	\(\frac{1}{4}\)
\(-1\)	\(\frac{1}{2}\)
\(0\)	\(1\)
\(1\)	\(2\)
\(2\)	\(4\)

(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.

As \(x \to \infty\text{,}\) \(f(x) \to -\infty\text{.}\)
As \(x \to \infty\text{,}\) \(f(x) \to -2\text{.}\)
As \(x \to \infty\text{,}\) \(f(x) \to 0\text{.}\)
As \(x \to \infty\text{,}\) \(f(x) \to 2\text{.}\)
As \(x \to \infty\text{,}\) \(f(x) \to \infty\text{.}\)

Answer.

(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.

As \(x \to-\infty\text{,}\) \(f(x) \to -\infty\text{.}\)
As \(x \to -\infty\text{,}\) \(f(x) \to -2\text{.}\)
As \(x \to -\infty\text{,}\) \(f(x) \to 0\text{.}\)
As \(x \to -\infty\text{,}\) \(f(x) \to 2\text{.}\)
As \(x \to -\infty\text{,}\) \(f(x) \to \infty\text{.}\)

Answer.

(d)

Complete the graph you started in Task 5.2.1.a, connecting the points and including the end behavior you’ve just determined.

Answer.

(e)

Does your graph seem to have any asymptotes?

No. There are no asymptotes.
There is a vertical asymptote but no horizontal one.
There is a horizontal asymptote but no vertical one.
The graph has both a horizontal and vertical asymptote.

Answer.

(f)

What the equation for each asymptote of \(f(x)\text{?}\) Select all that apply.

There are no asymptotes.
\(\displaystyle x=0\)
\(\displaystyle x=3\)
\(\displaystyle y=0\)
\(\displaystyle y=3\)

Answer.

(g)

Find the domain and range of \(f(x)\text{.}\) Write your answers using interval notation.

Answer.

Domain: \((-\infty,\infty)\text{,}\) Range: \((0, \infty)\)

(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.

Answer.

Increasing: \((-\infty,\infty)\text{,}\) Decreasing: nowhere

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 \((0,\infty)\text{.}\)
It is an exponential growth function; that is it is increasing on \((-\infty,\infty)\text{.}\)
There is a horizontal asymptote at \(y=0\text{.}\) There is no vertical asymptote.
There is a \(y\)-intercept at \((0,1)\text{.}\) There is no \(x\)-intercept.

Activity 5.2.3.

Consider the function \(g(x)=\left(\frac{1}{2}\right)^x\text{.}\)

(a)

Fill in the table of values for \(g(x)\text{.}\) Then plot the points on a graph.

\(x\)	\(g(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)

Answer.

\(x\)	\(g(x)\)
\(-2\)	\(4\)
\(-1\)	\(2\)
\(0\)	\(1\)
\(1\)	\(\frac{1}{2}\)
\(2\)	\(\frac{1}{4}\)

(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.

As \(x \to \infty\text{,}\) \(g(x) \to -\infty\text{.}\)
As \(x \to \infty\text{,}\) \(g(x) \to -2\text{.}\)
As \(x \to \infty\text{,}\) \(g(x) \to 0\text{.}\)
As \(x \to \infty\text{,}\) \(g(x) \to 2\text{.}\)
As \(x \to \infty\text{,}\) \(g(x) \to \infty\text{.}\)

Answer.

(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.

As \(x \to-\infty\text{,}\) \(g(x) \to -\infty\text{.}\)
As \(x \to -\infty\text{,}\) \(g(x) \to -2\text{.}\)
As \(x \to -\infty\text{,}\) \(g(x) \to 0\text{.}\)
As \(x \to -\infty\text{,}\) \(g(x) \to 2\text{.}\)
As \(x \to -\infty\text{,}\) \(g(x) \to \infty\text{.}\)

Answer.

(d)

Complete the graph you started in Task 5.2.3.a, connecting the points and including the end behavior you’ve just determined.

Answer.

(e)

What are the equations of the asymptote(s) of the graph?

Answer.

\(y=0\)

(f)

Find the domain and range of \(f(x)\text{.}\) Write your answers using interval notation.

Answer.

Domain: \((-\infty,\infty)\text{,}\) Range: \((0, \infty)\)

(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.

Answer.

Increasing: nowhere, Decreasing: \((-\infty,\infty)\)

Activity 5.2.4.

Consider the two exponential functions we’ve just graphed: \(f(x)=2^x\) and \(g(x)=\left(\frac{1}{2}\right)^x\text{.}\)

(a)

How are the graphs of \(f(x)\) and \(g(x)\) similar?

Answer.

Answers could include basic shape, asymptote, domain, range.

(b)

How are the graphs of \(f(x)\) and \(g(x)\) different?

Answer.

Answers could include reflection over \(y\)-axis, one is increasing, one is decreasing.

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.

The graph of an exponential function \(f(x)=b^x\) has the following characteristics:

Its domain is \((-\infty,\infty)\) and its range is \((0,\infty)\text{.}\)
Remember, exponential functions are only defined when \(b>0\) and \(b\neq 1\) as we saw in Definition 5.1.4.
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.
There is a horizontal asymptote at \(y=0\text{.}\) There is no vertical asymptote.
There is a \(y\)-intercept at \((0,1)\text{.}\) There is no \(x\)-intercept.

Activity 5.2.6.

Let’s look at a third exponential function, \(h(x)=2^{-x}\text{.}\)

(a)

Before plotting any points or graphing, what do you think the graph might look like? What sort of characteristics might it have?

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!

(b)

Fill in the table of values for \(h(x)\text{.}\) Then plot the points on a graph.

\(x\)	\(h(x)\)
\(-2\)
\(-1\)
\(0\)
\(1\)
\(2\)

Answer.

\(x\)	\(h(x)\)
\(-2\)	\(4\)
\(-1\)	\(2\)
\(0\)	\(1\)
\(1\)	\(\frac{1}{2}\)
\(2\)	\(\frac{1}{4}\)

(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.

Answer.

\(2^{-x} = (2^{-1})^x = \left(\frac{1}{2} \right)^x \)

(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 \(h(x)=2^{-x}\text{?}\)

A vertical stretch.
A horizontal stretch.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.

Answer.

Remark 5.2.7.

For a reminder of transformations, see Section 2.4 and the following definitions:

Activity 5.2.8.

Let \(f(x)=4^{x}\text{.}\)

(a)

Graph \(f(x)\text{.}\)

Answer.

(b)

Match the following functions to their graphs.

\(\displaystyle g(x)= -4^x \)
\(\displaystyle h(x)= 4^{-x} \)
\(\displaystyle j(x)= 4^{x+1} \)
\(\displaystyle k(x)= 4^x +1 \)

Answer.

\(g(x)= -4^x \) is D
\(h(x)= 4^{-x} \) is B
\(j(x)= 4^{x+1} \) is C
\(k(x)= 4^x +1 \) is A

(c)

Find the domain, range, and equation of the asymptote for the parent function \(\left(f(x)\right)\) and each of the four transformations \(\left(g(x), h(x), j(x), \text{ and } k(x)\right)\text{.}\)

Answer.

\(f(x)\text{:}\)

Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y=0\)

\(g(x)\text{:}\)

Domain: \((-\infty,\infty)\)
Range: \((-\infty,0)\)
Asymptote: \(y=0\)

\(h(x)\text{:}\)

Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y=0\)

\(j(x)\text{:}\)

Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y=0\)

\(k(x)\text{:}\)

Domain: \((-\infty,\infty)\)
Range: \((1,\infty)\)
Asymptote: \(y=1\)

(d)

Which of the transformations affected the domain of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

(e)

Which of the transformations affected the range of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

A and C

(f)

Which of the transformations affected the asymptote of the exponential function? Select all that apply.

A vertical shift.
A horizontal shift.
A reflection over the \(x\)-axis.
A reflection over the \(y\)-axis.
None of these.

Answer.

Activity 5.2.9.

Consider the function \(f(x)=e^{x}\text{.}\)

(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.)

Answer.

Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y=0\)

(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.

Answer.

Transformation: shift right 2
Domain: \((-\infty,\infty)\)
Range: \((0,\infty)\)
Asymptote: \(y=0\)

(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.

Answer.

Transformations: vertical stretch of 3, reflection over \(x\)-axis
Domain: \((-\infty,\infty)\)
Range: \((-\infty,0)\)
Asymptote: \(y=0\)

(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.

Answer.

Transformations: reflection over \(y\)-axis, shift down 4
Domain: \((-\infty,\infty)\)
Range: \((-4,\infty)\)
Asymptote: \(y=-4\)

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.