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Section 3.3 Graphs of Exponential Functions

In the previous section we listed some properties of exponential functions without much justification. In this section we will discuss why these properties hold, and investigate the long-run behavior of exponential functions.

Subsection 3.3.1 Graphs of Exponential Functions

The graphs of exponential functions have two characteristic shapes, depending on whether the base, \(b\text{,}\) is greater than \(1\) or less than \(1\text{.}\) As typical examples, consider the graphs of \(f (x) = 2^x\) and \(g(x) =\left(\dfrac{1}{2}\right)^x\) shown in Figure 3.17. Some values for \(f\) and \(g\) are recorded in Tables 3.15 and 3.16.
\(x\) \(f(x)\)
\(-3\) \(\frac{1}{8}\)
\(-2\) \(\frac{1}{4}\)
\(-1\) \(\frac{1}{2}\)
\(0\) \(1\)
\(1\) \(2\)
\(2\) \(4\)
\(3\) \(8\)
Table 3.15.
\(x\) \(g(x)\)
\(-3\) \(8\)
\(-2\) \(4\)
\(-1\) \(2\)
\(0\) \(1\)
\(1\) \(\frac{1}{2}\)
\(2\) \(\frac{1}{4}\)
\(3\) \(\frac{1}{8}\)
Table 3.16.
described in detail following the image
graphs of exponential growth and decay
Figure 3.17.
Notice that \(f (x) = 2^x\) is an increasing function and \(g(x) = \left(\dfrac{1}{2}\right)^x\) is a decreasing function. In general, exponential functions have the following properties.

Properties of Exponential Functions, \(f(x) = ab^x\text{,}\) \(a \gt 0\).

  1. Domain: all real numbers; \((-\infty,\infty)\text{.}\)
  2. Range: all positive numbers; \((0,\infty)\text{.}\)
  3. If \(b \gt 1\text{,}\) the function is increasing and said to have exponential growth ;
    if \(0 \lt b \lt 1\text{,}\) the function is decreasing and said to have exponential decay .
  4. The \(y\)-intercept is \((0, a)\text{.}\) There is no \(x\)-intercept.
In Table 3.15 you can see that as the \(x\)-values decrease toward negative infinity, the corresponding \(y\)-values decrease toward zero. As a result, the graph of \(f\) decreases toward the \(x\)-axis as we move to the left. Thus, the \(x\)-axis, \(y=0 \text{,}\) is a horizontal asymptote for exponential functions no matter what value \(b\text{,}\) the growth factor, happens to be. We can see this illustrated in Figure 3.17, since the \(y\)-values of each exponential get closer to zero. (For more info on horizontal asymptotes, see Appendix B.)
For exponential functions with \(0 \lt b \lt 1\text{,}\) this takes place close to the positive \(x\)-axis, as illustrated in Figure 3.17b. For exponential functions with \(1 \lt b\text{,}\) this takes place close to the negative \(x\)-axis, as illustrated in Figure 3.17a.
In Example 3.18, we compare two increasing exponential functions. The larger the value of the base, \(b\text{,}\) the faster the function grows. In this example, both functions have \(a = 1\text{.}\)

Example 3.18.

Compare the graphs of \(f (x) = 3^x\) and \(g(x) = 4^x\text{.}\)
Solution.
We evaluate each function for several convenient values, as shown in Table 3.19.
Plot the points for each function and connect them with smooth curves. For positive \(x\)-values, \(g(x)\) is always larger than \(f (x)\text{,}\) and is increasing more rapidly. In Figure 3.20, \(g(x) = 4^x\) climbs more rapidly than \(f (x) = 3^x\text{.}\) Both graphs cross the \(y\)-axis at (0, 1).
\(x\) \(f(x)\) \(g(x)\)
\(-2\) \(\dfrac{1}{9}\) \(\dfrac{1}{16}\)
\(-1\) \(\dfrac{1}{3}\) \(\dfrac{1}{4}\)
\(0\) \(1\) \(1\)
\(1\) \(3\) \(4\)
\(2\) \(9\) \(16\)
Table 3.19.
described in detail following the image
two exponential growth functions
Figure 3.20.
For decreasing exponential functions, those with bases between \(0\) and \(1\text{,}\) the smaller the base, the more steeply the graph decreases. For example, compare the graphs of \(p(x) = 0.8^x\) and \(q(x) = 0.5^x\) shown in Figure 3.21.
described in detail following the image
two exponential decay functions
Figure 3.21.

Example 3.22.

  1. State the ranges of the functions \(f\) and \(g\) in Figure 3.20 from Example 3.18 on the domain \([-2, 2]\text{.}\)
  2. State the ranges of the functions \(p\) and \(q\) shown in Figure 3.21 on the domain \([-2, 2]\text{.}\) Round your answers to two decimal places.
Solution.
  1. On the domain \([-2,2]\text{,}\) the smallest values for both \(f\) and \(g\) occur when \(x=-2\text{,}\) and the largest values occur when \(x=2\) (since both of these functions are increasing). Since
    \begin{equation*} \begin{aligned} f(-2) \amp= 3^{-2}=\frac{1}{9} \\ f(2) \amp= 3^2=9 \\ g(-2) \amp= 4^{-2}=\frac{1}{16} \\ g(2) \amp= 4^2= 16, \end{aligned} \end{equation*}
    the range of \(f\) on the domain \([-2,2]\) is \(\left[\frac{1}{9},9\right]\text{;}\) the range of \(g\) on the domain \([-2,2]\) is \(\left[\frac{1}{16},16\right]\text{.}\)
  2. On the domain \([-2,2]\text{,}\) the smallest values for both \(p\) and \(q\) occurs when \(x=2\text{,}\) and the largest values occurs when \(x=-2\) (since both of these functions are decreasing). Since
    \begin{equation*} \begin{aligned} p(-2) \amp= 0.5^{-2}=4 \\ p(2) \amp= 0.5^2=0.25 \\ q(-2) \amp= 0.8^{-2}\approx1.56 \\ q(2) \amp= 0.8^2= 0.64, \end{aligned} \end{equation*}
    the range of \(p\) on the domain \([-2,2]\) is \(\left[0.25,4\right]\text{;}\) the range of \(q\) on the domain \([-2,2]\) is \(\left[0.64,1.5625\right]\text{.}\)

Exercises 3.3.2 Exercises

1. Comparing Exponential Functions.

Let \(f(x)=5(1.06)^x\) and \(g(x)=7(1.01)^x\text{.}\)
a) Which function has a larger growth rate?
b) Which function has a larger \(y\)-intercept?
c) Using your answers from parts (a) and (b), will the graphs of \(f(x)\) and \(g(x)\) cross in the first quadrant or the second quadrant?
Hint.
a) Recall the growth rate is given by \(b-1\) in an exponential function \(y=a(b)^t\text{.}\) Find the growth rate for each formula and compare.
b) Recall the initial value of an exponential function is given by \(a\) in \(y=a(b)^t\text{.}\) Compare the initial values in each formula.
c) Think about which function starts off below the other and which one is growing faster. It may be helpful to sketch a graph of the two.

2. Linear vs. Exponential Functions: Table.

Decide whether the function \(f(x)\) could be a linear or exponential function. Then, find a possible formula for \(f(x)\text{.}\)
\(x\) \(0\) \(1\) \(2\) \(3\) \(4\)
\(f(x)\) 6 6.6 7.26 7.986 8.7846
Values of \(f(x)\)
\(f(x) =\)
Hint.
You should determine if you add a constant for each increase by 1 in \(x\) or if you multiply by a constant for each increase by 1 in \(x\text{.}\)
A linear function can be verified by checking that there’s a constant average rate of change in the table:
\begin{equation*} \frac{f(b)-f(a)}{b-a}. \end{equation*}
An exponential function can be verified by checking that there’s a constant ratio between values in the table.
Recall the formulas for a linear and an exponential function:
\begin{equation*} \text{Linear: } y=mt+b \end{equation*}
\begin{equation*} \text{Exponential: } y=a(b)^t. \end{equation*}
Note the "b"s in each equation represent different values!

3. Calculating Exponential Growth.

Suppose a certain population of bacteria is known to grow by 16 percent each day. If a lab has 1200 bacteria on May 1, how many bacteria will there be 25 days from then?
Hint.
Recall the formula for an exponential function
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know \(b=1+r\text{,}\) where \(r\) is the growth rate.
Use this information to write an equation that gives the population of bacteria as a function of \(t\) days. Then you’ll want to evaluate this equation.

4. Writing and Evaluating Exponential Functions.

Suppose that the number of new cases of a certain disease \(t\) years after 1950, \(I(t)\text{,}\) is reduced by 2 percent per year.
a) If there were 5000 new cases of the disease in 1950, write a formula for \(I(t)\text{.}\)
\(I(t) =\)
b) Based on your model from part (a), approximately how many new cases will there be in 2040?
cases
Hint.
a) Recall the formula for an exponential function
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know \(b=1+r\text{,}\) where \(r\) is the growth rate. Note here that the number of cases is decreasing, so \(r\) should be negative.
b) Once you have a formula, you can evaluate it for \(t=90\text{.}\)

5. Writing and Evaluating Exponential Functions.

The Emerald City’s population density shrunk throughout the 1970s. In 1970, the population density was approximately 7710 people per square mile, and it was shrinking at a rate of 0.8 percent per year.
a) If this trend had continued, find a formula for the population density, \(D(t)\text{,}\) of the Emerald City \(t\) years after 1970.
\(D(t) =\)
b) Assuming this trend continues, predict the population density of the Emerald City in 1980 using your function from part (a).
people per square mile
Hint.
a) Recall the formula for an exponential function
\begin{equation*} y=a(b)^t, \end{equation*}
where \(a\) is the initial value and \(b\) is the growth factor. We know \(b=1+r\text{,}\) where \(r\) is the growth rate. Note here that the population density is decreasing, so \(r\) should be negative.
b) Once you have a formula, you can evaluate it for \(t=10\text{.}\)

6. End Behavior of Exponential Functions.

Suppose that the percentage of children not vaccinated against the disease Dermatitis is given by the equation \(V(t)=3+97(0.84)^t\text{,}\) where \(t\) is in years since 1950. As \(t\) goes to infinity, what happens to \(V(t)\text{?}\) What does this mean in practical terms? Select the answer that best fits the situation.
  • As \(t\) becomes large, \(V(t)\) gets closer and closer to zero. That is, eventually every child receives the vaccination.
  • As \(t\) becomes large, \(V(t)\) becomes very large. That is, eventually the number of unvaccinated children becomes very large.
  • As \(t\) becomes large, \(V(t)\) gets closer and closer to 3. That is, eventually almost every child receives the vaccination.
  • As \(t\) becomes large, \(V(t)\) gets closer and closer to 97. That is, eventually almost every child receives the vaccination.
Hint.
Try to plug in a very large positive number for \(t\) to symbolize \(t\) going to infinity. You may want to plug in increasing numbers to see what values the function \(V(t)\) is getting closer and closer to.

7. Comparing Exponential Functions.

Let \(f(x) = 11(1.47)^x\) and \(g(x) = 11(1.59)^x\text{.}\)
a) Which function is growing faster?
b) Graph both \(f(x)\) and \(g(x)\) on your own piece of paper, then compute the average rate of change for both functions on the intervals \([0,5]\) and \([5,10]\text{.}\)
What is the average rate of change of:
\(f(x)\) on the interval \([0,5]\text{:}\)
\(f(x)\) on the interval \([5,10]\)
\(g(x)\) on the interval \([0,5]\)
\(g(x)\) on the interval \([5,10]\)
c) Which function has a greater average rate of change on these intervals? Be sure to decide if this agrees with what you expected.
Hint.
a) Compare growth factors or growth rates.
b) The formula for the average rate of change over an interval for \(f(x)\) is given by \(\frac{f(b) - f(a)}{b-a}\text{,}\) where \(a\) is the left endpoint of the interval and \(b\) is the right endpoint of the interval. Calculate the average rate of change on each interval using this information and the given functions.
c) Compare the average rates of change calculated in part b). Compare this answer to part a).

8. Piecewise Functions.

Use the function \(g(x)\) as defined below to answer parts (a)-(d). \(\begin{align*} g(x) = \begin{cases} x+6, \amp x\lt 0\\ 3, \amp x=0\\ 6(0.5)^x, \amp x>0 \end{cases} \end{align*}\)
a) Which of the following is an accurate sketch a graph of \(g(x)\) on the interval \([-5,5]\text{?}\)
A B
C D
(Click on a graph to enlarge it)
b) What are the domain and range of \(g(x)\text{?}\)
  • Domain: \((-\infty, \infty)\text{,}\) Range: \((-\infty, \infty)\)
  • Domain: \((-\infty, \infty)\text{,}\) Range: \((-\infty, 0) \cup (0, 6)\)
  • Domain: \([-\infty, \infty]\text{,}\) Range: \((-\infty, 6]\)
  • Domain: \((-\infty, \infty)\text{,}\) Range: \((-\infty, 6)\)
c) Over what intervals is \(g(x)\) increasing? Decreasing?
  • Increasing: \((-\infty, 0]\text{;}\) Decreasing: \((0, \infty)\)
  • Increasing: \((-\infty, 6)\text{;}\) Decreasing: \((6, 0)\)
  • Increasing: \((-\infty, 0)\text{;}\) Decreasing: \([0, \infty]\)
  • Increasing: \((-\infty, 0)\text{;}\) Decreasing: \((0, \infty)\)
d) State the long-run behavior of \(g(x)\) as \(x \to \infty\text{,}\) and give the equation(s) of any horizontal asymptote(s).
Long-run behavior:
  • As \(x\) goes to \(\infty\text{,}\) \(g(x)\) approaches \(6\text{.}\)
  • As \(x\) goes to \(\infty\text{,}\) \(g(x)\) approaches 0.
  • As \(x\) goes to \(\infty\text{,}\) \(g(x)\) approaches \(- \infty\)
  • As \(x\) goes to \(\infty\text{,}\) \(g(x)\) approaches \(\infty\text{.}\)
Horizontal asymptote(s):
Hint.
a) We are able to identify the correct graph of \(g(x)\) by examining the endpoints and the domains of the pieces.
b) Use the graph from part a) to help you figure this out. Recall the domain is the set (interval) of all possible input values and the range is the set (interval) of associated output values.
c) Use the graph from part a) to help you figure this out. Recall a function is increasing if it looks like it’s "going up" as you move from left to right and a function is decreasing if it looks like it’s "going down" as you move from left to right.
Also, remember that we don’t include the endpoints when discussing increasing/decreasing.
d) Think about what happens to the values of the function as you plug in larger and larger values of \(x\text{.}\) In other words, what value is the function getting closer to as you move to the right on the input axis?
If the function is approaching a constant number (not infinity), the function has a horizontal asymptote. Horizontal asymptotes are lines of the form \(y=k\) for some constant \(k\text{.}\)
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