6.2 Graphs of Exponential Functions
As we discussed in the previous section, exponential functions are used for many real-world applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a real-world situation gives us a method for making predictions. Most of the time, however, the equation itself is not enough. We learn a lot about things by seeing their pictorial representations, and that is exactly why graphing exponential equations is a powerful tool. It gives us another layer of insight for predicting future events.
6.2.1 Graphing Exponential Functions
Before we begin graphing, it is helpful to review the behavior of exponential growth. Recall the table of values for a function of the form \(f(x) = b^{x}\) whose base is greater than one. We’ll use the function \(f(x) = 2^{x}.\) Observe how the output values in Table 1 change as the input increases by \(1.\)
| \(x\) | \(- 3\) | \(- 2\) | \(- 1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(f(x) = 2^{x}\) | \(\frac{1}{8}\) | \(\frac{1}{4}\) | \(\frac{1}{2}\) | \(1\) | \(2\) | \(4\) | \(8\) |
Table 1
Each output value is the product of the previous output and the base, \(2.\) We call the base \(2\) the constant ratio. In fact, for any exponential function with the form \(f(x) = ab^{x},\) \(b\) is the constant ratio of the function. This means that as the input increases by 1, the output value will be the product of the base and the previous output, regardless of the value of \(a.\)
Notice from the table that
- the output values are positive for all values of \(x;\)
- as \(x\) increases, the output values increase without bound; and
- as \(x\) decreases, the output values grow smaller, approaching zero.
Figure 1 shows the exponential growth function \(f(x) = 2^{x}.\)
Figure 1 Notice that the graph gets close to the \(x\)-axis, but never touches it.
The domain of \(f(x) = 2^{x}\) is all real numbers, the range is \(\left( {0,\infty} \right),\) and the horizontal asymptote is \(y = 0.\)
To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form \(f(x) = b^{x}\) whose base is between zero and one. We’ll use the function \(g(x) = \left( \frac{1}{2} \right)^{x}.\) Observe how the output values in Table 2 change as the input increases by \(1.\)
| \(x\) | \(-3\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \({g(x) =}\left( \frac{1}{2} \right)^{x}\) | \(8\) | \(4\) | \(2\) | \(1\) | \(\frac{1}{2}\) | \(\frac{1}{4}\) | \(\frac{1}{8}\) |
Table 2
Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio \(\frac{1}{2}.\)
Notice from the table that
- the output values are positive for all values of \(x;\)
- as \(x\) increases, the output values grow smaller, approaching zero; and
- as \(x\) decreases, the output values grow without bound.
Figure 2 shows the exponential decay function, \(g(x) = \left( \frac{1}{2} \right)^{x}.\)
Figure 2
The domain of \(g(x) = \left( \frac{1}{2} \right)^{x}\) is all real numbers, the range is \(\left( {0,\infty} \right),\) and the horizontal asymptote is \(y = 0.\)
An exponential function with the form \(f(x) = b^{x},\) \(b > 0,\) \(b \neq 1,\) has these characteristics:
- one-to-one function
- horizontal asymptote: \(y = 0\)
- domain: \((–\infty,~\infty)\)
- range: \((0,\infty)\)
- \(x\)-intercept: none
- \(y\)-intercept: \(\left( {0,1} \right)\)
- increasing if \(b > 1\)
- decreasing if \(b < 1\)
Figure 3 compares the graphs of exponential growth and decay functions.
Figure 3
Sketch a graph of \(f(x) = 0.25^{x}.\) State the domain, range, and asymptote.
Solution (click to reveal)
Before graphing, identify the behavior and create a table of points for the graph.
Since \(b = 0.25\) is between zero and one, we know the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote \(y = 0.\)
Create a table of points as in Table 3.
| \(x\) | \(- 3\) | \(- 2\) | \(- 1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \(f(x) = 0.25^{x}\) | \(64\) | \(16\) | \(4\) | \(1\) | \(0.25\) | \(0.0625\) | \(0.015625\) |
Table 3
- Plot the \(y\)-intercept, \(\left( {0,1} \right),\) along with two other points. We can use \(\left( {- 1,4} \right)\) and \(\left( {1,0.25} \right).\)
Draw a smooth curve connecting the points as in Figure 4.
Figure 4
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {0,\infty} \right);\) the horizontal asymptote is \(y = 0.\)
Sketch the graph of \(f(x) = 4^{x}.\) State the domain, range, and asymptote.
Solution (click to reveal)
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {0,\infty} \right);\) the horizontal asymptote is \(y = 0.\)
6.2.2 Graphing Transformations of Exponential Functions
Transformations of exponential graphs behave similarly to those of other functions. Just as with other parent functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the parent function \(f(x) = b^{x}\) without loss of shape. For instance, just as the quadratic function maintains its parabolic shape when shifted, reflected, stretched, or compressed, the exponential function also maintains its general shape regardless of the transformations applied.
Graphing a Vertical Shift
The first transformation occurs when we add a constant \(d\) to the parent function \(f(x) = b^{x},\) giving us a vertical shift \(d\) units in the same direction as the sign. For example, if we begin by graphing a parent function, \(f(x) = 2^{x},\) we can then graph two vertical shifts alongside it, using \(d = 3:\) the upward shift, \(g(x) = 2^{x} + 3\) and the downward shift, \(h(x) = 2^{x} - 3.\) Both vertical shifts are shown in Figure 5.
Figure 5
Observe the results of shifting \(f(x) = 2^{x}\) vertically:
The domain, \(\left( {- \infty,\infty} \right)\) remains unchanged.
When the function is shifted up \(3\) units to \(g(x) = 2^{x} + 3:\)
The \(y\)-intercept shifts up \(3\) units to \(\left( {0,4} \right).\)
The asymptote shifts up \(3\) units to \(y = 3.\)
The range becomes \(\left( {3,\infty} \right).\)
When the function is shifted down \(3\) units to \(h(x) = 2^{x} - 3:\)
The \(y\)-intercept shifts down \(3\) units to \(\left( {0, - 2} \right).\)
The asymptote also shifts down \(3\) units to \(y = - 3.\)
The range becomes \(\left( {- 3,\infty} \right).\)
Graphing a Horizontal Shift
The next transformation occurs when we add a constant \(c\) to the input of the parent function \(f(x) = b^{x},\) giving us a horizontal shift \(c\) units in the opposite direction of the sign. For example, if we begin by graphing the parent function \(f(x) = 2^{x},\) we can then graph two horizontal shifts alongside it, using \(c = 3:\) the shift left, \(g(x) = 2^{x + 3},\) and the shift right, \(h(x) = 2^{x - 3}.\) Both horizontal shifts are shown in Figure 6.
Figure 6
Observe the results of shifting \(f(x) = 2^{x}\) horizontally:
The domain, \(\left( {- \infty,\infty} \right),\) remains unchanged.
The asymptote, \(y = 0,\) remains unchanged.
The \(y\)-intercept shifts such that:
When the function is shifted left \(3\) units to \(g(x) = 2^{x + 3},\) the \(y\)-intercept becomes \(\left( {0,8} \right).\) This is because \(2^{x + 3} = (8)2^{x},\) so the initial value of the function is \(8.\)
When the function is shifted right \(3\) units to \(h(x) = 2^{x - 3},\) the \(y\)-intercept becomes \(\left( {0,\frac{1}{8}} \right).\) Again, see that \(2^{x - 3} = \left( \frac{1}{8} \right)2^{x},\) so the initial value of the function is \(\frac{1}{8}.\)
Graph \(f(x) = 2^{x + 1} - 3.\) State the domain, range, and asymptote.
Solution (click to reveal)
We have an exponential equation of the form \(f(x) = b^{x + c} + d,\) with \(b = 2,\) \(c = 1,\) and \(d = -3.\)
Draw the horizontal asymptote \(y = d\) , so draw \(y = -3.\)
Identify the shift as \(\left( {- c,d} \right),\) so the shift is \(\left( {- 1,-3} \right).\)
Shift the graph of \(f(x) = b^{x}\) left 1 units and down 3 units.
Figure 7
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {- 3,\infty} \right);\) the horizontal asymptote is \(y = -3.\)
Graph \(f(x) = 2^{x - 1} + 3.\) State domain, range, and asymptote.
Solution (click to reveal)
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {3,\infty} \right);\) the horizontal asymptote is \(y = 3.\)
Graphing a Stretch or Compression
While horizontal and vertical shifts involve adding constants to the input or to the function itself, a stretch or compression occurs when we multiply the parent function \(f(x) = b^{x}\) by a constant \(\left| a \middle| > 0. \right.\) For example, if we begin by graphing the parent function \(f(x) = 2^{x},\) we can then graph the stretch, using \(a = 3,\) to get \(g(x) = 3(2)^{x}\) as shown on the left in Figure 8, and the compression, using \(a = \frac{1}{3},\) to get \(h(x) = \frac{1}{3}(2)^{x}\) as shown on the right in Figure 8.
Figure 8 (a) \(g(x) = 3(2)^{x}\) stretches the graph of \(f(x) = 2^{x}\) vertically by a factor of \(3.\) (b) \(h(x) = \frac{1}{3}(2)^{x}\) compresses the graph of \(f(x) = 2^{x}\) vertically by a factor of \(\frac{1}{3}.\)
Graphing the Stretch of an Exponential Function
Sketch a graph of \(f(x) = 4\left( \frac{1}{2} \right)^{x}.\) State the domain, range, and asymptote.
Solution (click to reveal)
Before graphing, identify the behavior and key points on the graph.
Since \(b = \frac{1}{2}\) is between zero and one, the left tail of the graph will increase without bound as \(x\) decreases, and the right tail will approach the \(x\)-axis as \(x\) increases.
Since \(a = 4,\) the graph of \(f(x) = \left( \frac{1}{2} \right)^{x}\) will be stretched by a factor of \(4.\)
Create a table of points as shown in Table 4.
| \(x\) | \(- 3\) | \(- 2\) | \(- 1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \({f(x)}{= 4}\left( \frac{1}{2} \right)^{x}\) | \(32\) | \(16\) | \(8\) | \(4\) | \(2\) | \(1\) | \(0.5\) |
Table 4
- Plot the \(y\)-intercept, \(\left( {0,4} \right),\) along with two other points. We can use \(\left( {- 1,8} \right)\) and \(\left( {1,2} \right).\)
Draw a smooth curve connecting the points, as shown in Figure 9.
Figure 9
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {0,\infty} \right);\) the horizontal asymptote is \(y = 0.\)
Sketch the graph of \(f(x) = \frac{1}{2}(4)^{x}.\) State the domain, range, and asymptote.
Solution (click to reveal)
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {0,\infty} \right);\) the horizontal asymptote is \(y = 0.\)
Graphing Reflections
In addition to shifting, compressing, and stretching a graph, we can also reflect it about the \(x\)-axis or the \(y\)-axis. When we multiply the parent function \(f(x) = b^{x}\) by \(-1,\) we get a reflection about the \(x\)-axis. When we multiply the input by \(-1,\) we get a reflection about the \(y\)-axis. For example, if we begin by graphing the parent function \(f(x) = 2^{x},\) we can then graph the two reflections alongside it. The reflection about the \(x\)-axis, \(g(x) = -2^{x},\) is shown on the left side of Figure 10, and the reflection about the \(y\)-axis \(h(x) = 2^{- x},\) is shown on the right side of Figure 10.
Figure 10 (a) \(g(x) = - 2^{x}\) reflects the graph of \(f(x) = 2^{x}\) about the x-axis. (b) \(g(x) = 2^{- x}\) reflects the graph of \(f(x) = 2^{x}\) about the \(y\)-axis.
Find and graph the equation for a function, \(g(x),\) that reflects \(f(x) = \left( \frac{1}{4} \right)^{x}\) about the \(x\)-axis. State its domain, range, and asymptote.
Solution (click to reveal)
Since we want to reflect the parent function \(f(x) = \left( \frac{1}{4} \right)^{x}\) about the \(x\)-axis, we multiply \(f(x)\) by \(- 1\) to get, \(g(x) = - \left( \frac{1}{4} \right)^{x}.\) Next we create a table of points as in Table 5.
| \(x\) | \(- 3\) | \(- 2\) | \(- 1\) | \(0\) | \(1\) | \(2\) | \(3\) |
| \({g(x) = -}\left( \frac{1}{4} \right)^{x}\) | \(- 64\) | \(- 16\) | \(- 4\) | \(- 1\) | \(- 0.25\) | \(- 0.0625\) | \(- 0.0156\) |
Table 5
Plot the \(y\)-intercept, \(\left( {0,-1} \right),\) along with two other points. We can use \(\left( {-1,-4} \right)\) and \(\left( {1,-0.25} \right).\)
Draw a smooth curve connecting the points:
Figure 11
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {- \infty,0} \right);\) the horizontal asymptote is \(y = 0.\)
Find and graph the equation for a function, \(g(x),\) that reflects \(f(x) = 1.25^{x}\) about the \(y\)-axis. State its domain, range, and asymptote.
Solution (click to reveal)
The domain is \(\left( {- \infty,\infty} \right);\) the range is \(\left( {0,\infty} \right);\) the horizontal asymptote is \(y = 0.\)
Summarizing Translations of the Exponential Function
Now that we have worked with each type of translation for the exponential function, we can summarize them in Table 6 to arrive at the general equation for translating exponential functions.
| Transformations of the Parent Function \(f(x) = b^{x}\) | |
|---|---|
| Transformation | Form |
Shift
|
\(f(x) = b^{x + c} + d\) |
Stretch and Compress
|
\(f(x) = ab^{x}\) |
| Reflect about the \(x\)-axis | \(f(x) = - b^{x}\) |
| Reflect about the \(y\)-axis | \(f(x) = b^{- x} = \left( \frac{1}{b} \right)^{x}\) |
| General equation for all transformations | \(f(x) = ab^{x + c} + d\) |
Table 6
Section Exercises
Verbal
1. What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?
Solution (click to reveal)
An asymptote is a line that the graph of a function approaches, as \(x\) either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the limit of the function’s values as the independent variable gets either extremely large or extremely small.
2. What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?
Algebraic
3. The graph of \(f(x) = 3^{x}\) is reflected about the \(y\)-axis and stretched vertically by a factor of \(4.\) What is the equation of the new function, \(g(x)?\) State its \(y\)-intercept, domain, and range.
Solution (click to reveal)
\(g(x) = 4(3)^{- x};\) \(y\)-intercept: \((0,4);\) Domain: all real numbers; Range: all real numbers greater than \(0.\)
4. The graph of \(f(x) = \left( \frac{1}{2} \right)^{- x}\) is reflected about the \(y\)-axis and compressed vertically by a factor of \(\frac{1}{5}.\) What is the equation of the new function, \(g(x)?\) State its \(y\)-intercept, domain, and range.
5. The graph of \(f(x) = 10^{x}\) is reflected about the \(x\)-axis and shifted upward \(7\) units. What is the equation of the new function, \(g(x)?\) State its \(y\)-intercept, domain, and range.
Solution (click to reveal)
\(g(x) = - 10^{x} + 7;\) \(y\)-intercept: \(\left( {0,6} \right);\) Domain: all real numbers; Range: all real numbers less than \(7.\)
6. The graph of \(f(x) = (1.68)^{x}\) is shifted right \(3\) units, stretched vertically by a factor of \(2,\) reflected about the \(x\)-axis, and then shifted downward \(3\) units. What is the equation of the new function, \(g(x)?\) State its \(y\)-intercept (to the nearest thousandth), domain, and range.
7. The graph of \(f(x) = - \frac{1}{2}\left( \frac{1}{4} \right)^{x - 2} + 4\) is shifted downward \(4\) units, and then shifted left \(2\) units, stretched vertically by a factor of \(4,\) and reflected about the \(x\)-axis. What is the equation of the new function, \(g(x)?\) State its \(y\)-intercept, domain, and range.
Solution (click to reveal)
\(g(x) = 2\left( \frac{1}{4} \right)^{x};\) \(y\)-intercept: \(\left( {0,\mspace{9mu}\text{2}} \right);\) Domain: all real numbers; Range: all real numbers greater than \(0.\)
Graphical
For the following exercises, graph the function and its reflection about the \(y\)-axis on the same axes, and give the \(y\)-intercept.
8. \(f(x) = 3\left( \frac{1}{2} \right)^{x}\)
9. \(g(x) = - 2(0.25)^{x}\)
Solution (click to reveal)
\(y\)-intercept: \((0, - 2)\)
10. \(h(x) = 6(1.75)^{- x}\)
For the following exercises, graph each set of functions on the same axes.
11. \(f(x) = 3\left( \frac{1}{4} \right)^{x},\) \(g(x) = 3(2)^{x},\) and \(h(x) = 3(4)^{x}\)
Solution (click to reveal)
12. \(f(x) = \frac{1}{4}(3)^{x},\) \(g(x) = 2(3)^{x},\) and \(h(x) = 4(3)^{x}\)
For the following exercises, match each function with one of the graphs in Figure 12.
Figure 12
13. \(f(x) = 2(0.69)^{x}\)
Solution (click to reveal)
B
14. \(f(x) = 2(1.28)^{x}\)
15. \(f(x) = 2(0.81)^{x}\)
Solution (click to reveal)
A
16. \(f(x) = 4(1.28)^{x}\)
17. \(f(x) = 2(1.59)^{x}\)
Solution (click to reveal)
E
18. \(f(x) = 4(0.69)^{x}\)
For the following exercises, use the graphs shown in Figure 13. All have the form \(f(x) = ab^{x}.\)
Figure 13
19. Which graph has the largest value for \(b?\)
Solution (click to reveal)
D
20. Which graph has the smallest value for \(b?\)
21. Which graph has the largest value for \(a?\)
Solution (click to reveal)
C
22. Which graph has the smallest value for \(a?\)
For the following exercises, graph the function and its reflection about the \(x\)-axis on the same axes.
23. \(f(x) = \frac{1}{2}(4)^{x}\)
Solution (click to reveal)
24. \(f(x) = 3(0.75)^{x} - 1\)
25. \(f(x) = - 4(2)^{x} + 2\)
Solution (click to reveal)
For the following exercises, graph the transformation of \(f(x) = 2^{x}.\) Give the horizontal asymptote, the domain, and the range.
26. \(f(x) = 2^{- x}\)
27. \(h(x) = 2^{x} + 3\)
Solution (click to reveal)
Horizontal asymptote: \(h(x) = 3;\operatorname{}\) Domain: all real numbers; Range: all real numbers strictly greater than \(3.\)
28. \(f(x) = 2^{x - 2}\)
For the following exercises, describe the end behavior of the graphs of the functions.
29. \(f(x) = - 5(4)^{x} - 1\)
Solution (click to reveal)
As \(x\rightarrow\infty\) , \(f(x)\rightarrow - \infty\) ;
As \(x\rightarrow - \infty\) , \(f(x)\rightarrow - 1\)
30. \(f(x) = 3\left( \frac{1}{2} \right)^{x} - 2\)
31. \(f(x) = 3(4)^{- x} + 2\)
Solution (click to reveal)
As \(x\rightarrow\infty\) , \(f(x)\rightarrow 2\) ;
As \(x\rightarrow - \infty\) , \(f(x)\rightarrow\infty\)
For the following exercises, start with the graph of \(f(x) = 4^{x}.\) Then write a function that results from the given transformation.
32. Shift \(f(x)\) 4 units upward
33. Shift \(f(x)\) 3 units downward
Solution (click to reveal)
\(f(x) = 4^{x} - 3\)
34. Shift \(f(x)\) 2 units left
35. Shift \(f(x)\) 5 units right
Solution (click to reveal)
\(f(x) = 4^{x - 5}\)
36. Reflect \(f(x)\) about the \(x\)-axis
37. Reflect \(f(x)\) about the \(y\)-axis
Solution (click to reveal)
\(f(x) = 4^{- x}\)
For the following exercises, each graph is a transformation of \(y = 2^{x}.\) Write an equation describing the transformation.
38.
39.
Solution (click to reveal)
\(y = - 2^{x} + 3\)
40.
For the following exercises, find an exponential equation for the graph.
41.
Solution (click to reveal)
\(y = - 2(3)^{x} + 7\)
42.
Numeric
For the following exercises, evaluate the exponential functions for the indicated value of \(x.\)
43. \(g(x) = \frac{1}{3}(7)^{x - 2}\) for \(g(6).\)
Solution (click to reveal)
\(g(6) = 800 + \frac{1}{3} \approx 800.3333\)
44. \(f(x) = 4{(2)}^{x - 1} - 2\) for \(f(5).\)
45. \(h(x) = - \frac{1}{2}\left( \frac{1}{2} \right)^{x} + 6\) for \(h( - 7).\)
Solution (click to reveal)
\(h( - 7) = - 58\)
Technology
For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.
46. \(- 50 = - \left( \frac{1}{2} \right)^{- x}\)
47. \(116 = \frac{1}{4}\left( \frac{1}{8} \right)^{x}\)
Solution (click to reveal)
\(x \approx - 2.953\)
48. \(12 = 2(3)^{x} + 1\)
49. \(5 = 3\left( \frac{1}{2} \right)^{x - 1} - 2\)
Solution (click to reveal)
\(x \approx - 0.222\)
50. \(- 30 = - 4(2)^{x + 2} + 2\)
Extensions
51. Explore and discuss the graphs of \(F(x) = (b)^{x}\) and \(G(x) = \left( \frac{1}{b} \right)^{x}.\) Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left( \frac{1}{b} \right)^{x}\) for any real number \(b > 0.\)
Solution (click to reveal)
The graph of \(G(x) = \left( \frac{1}{b} \right)^{x}\) is the refelction about the \(y\)-axis of the graph of \(F(x) = b^{x};\) For any real number \(b > 0\) and function \(f(x) = b^{x},\) the graph of \(\left( \frac{1}{b} \right)^{x}\) is the the reflection about the \(y\)-axis, \(F( - x).\)
52. Prove the conjecture made in the previous exercise.
53. Explore and discuss the graphs of \(f(x) = 4^{x},\) \(g(x) = 4^{x - 2},\) and \(h(x) = \left( \frac{1}{16} \right)4^{x}.\) Then make a conjecture about the relationship between the graphs of the functions \(b^{x}\) and \(\left( \frac{1}{b^{n}} \right)b^{x}\) for any real number n and real number \(b > 0.\)
Solution (click to reveal)
The graphs of \(g(x)\) and \(h(x)\) are the same and are a horizontal shift to the right of the graph of \(f(x);\) For any real number \(n\), real number \(b > 0,\) and function \(f(x) = b^{x},\) the graph of \(\left( \frac{1}{b^{n}} \right)b^{x}\) is the horizontal shift \(f(x - n).\)
54. Prove the conjecture made in the previous exercise.