Graphs of Exponential Functions

Learning Objectives

Graph exponential functions (IA 10.2.1).
Function transformations (exponential) (CA 3.5.1-3.5.5).

Objective 1: Graph exponential functions (IA 10.2.1).

Example 1

Graph exponential functions.

On the same coordinate system graph $f (x) = 2^{x}$ and $g (x) = 2^{x + 1} .$

Solution

We will use point plotting to graph the functions.

This figure shows two curves. The first curve is marked in blue and passes through the points (negative 1, 1 over 2), (0, 1) and (1, 2). The second curve is marked in red and passes through the points (negative 1, 1), (0, 2) and (1, 4).
Looking at the graphs of the functions $f (x) = 2^{x}$ and $g (x) = 2^{x + 1}$ above, we see that adding one in the exponent caused a horizontal shift of one unit to the left. We can use this pattern to graph other functions using horizontal shifts.

On the same coordinate system graph $f (x) = 3^{x}$ and $g (x) = 3^{x} - 2 .$

Solution

We will use point plotting to graph the functions.

This figure shows two curves. The first curve is marked in blue and passes through the points (negative 1, 1 over 3), (0, 1), and (1, 3). The second curve is marked in red and passes through the points (negative 1, negative 5 over 3), (0, negative 1), and (1, 1).
Looking at the graphs of the functions $f (x) = 3^{x}$ and $g (x) = 3^{x} - 2$ , we see that subtracting 2 caused vertical shift of down two units. Notice that the horizontal asymptote also shifted down 2 units. We can use this pattern to help graph other functions with a vertical shift.

Practice Makes Perfect

On the same coordinate system graph $f (x) = 3^{x}$ and $g (x) = 3^{x -1} .$ A Cartesian coordinate system is shown with a grid. The x-axis ranges from -5 to 5, labeled at integer intervals. The y-axis ranges from -4 to 12, labeled at even integer intervals from -4 to 12. Both axes have arrows at their ends, and the labels 'x' and 'y' are present for their respective axes.

On the same coordinate system graph $f (x) = 3^{x}$ and $g (x) = 3^{x} + 1 .$ A Cartesian coordinate system is shown with a grid. The x-axis ranges from -5 to 5, labeled at integer intervals. The y-axis ranges from -4 to 12, labeled at even integer intervals from -4 to 12. Both axes have arrows at their ends, and the labels 'x' and 'y' are present for their respective axes.

Objective 2: Function transformations (exponential). (CA 3.5.1-3.5.5)

Vertical and Horizontal Shifts:

Given a function

f (x)

, a new function

g (x) = f (x) + k

where

k

is a constant, is a vertical shift of the function

f (x)

. All the output values change by k units. If k is a positive, the graph will shift up. If k is negative, the graph will shift down.

Given a function $f (x)$ , a new function $g (x) = f (x - h)$ , where h is a constant, is a horizontal shift of the function $f (x)$ . If h is positive, the graph will shift right. If h is negative, the graph will shift left.

Example 2

Function transformations (exponential).

Graph $f (x) = 3^{x + 2} - 3$

Solution

Make a table for $f (x) = 3^{x}$
Add a column on the left for $x + 2$ , by subtracting 2 from all the input values
Add a column on the right by subtracting 3 from all the y-value
Two outside columns have the points for the new graph

A graph shows two exponential functions, y=3^x (red) and y=3^(x+2)-3 (blue), along with a table of corresponding x and y values for each. The blue function is a transformation of the red function.

Practice Makes Perfect

Function transformations (exponential).

Graph $f (x) = 2^{x - 3} - 1$ A Cartesian coordinate system is shown with a grid. The x-axis ranges from -5 to 5, labeled at integer intervals. The y-axis ranges from -4 to 12, labeled at even integer intervals from -4 to 12. Both axes have arrows at their ends, and the labels 'x' and 'y' are present for their respective axes.

ⓐ Given $f (x) = 3^{x}$ , reflect it about y-axis and write an equation of a new function below.
ⓑ Given $f (x) = 3^{x}$ , reflect it about x-axis and write an equation of a new function below.
ⓒ Given $f (x) = 3^{x}$ , shift the graph up 4 units and write an equation of a new function below.
ⓓ Graph the equations found in parts a, b, and c on the coordinate system provided and check your work using a graphing utility.

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 $1.$

Table 1 Two rows and eight columns. The first row is labeled, “x”, and the second row is labeled, “f(x)=2^x”. Reading the columns as ordered pairs, we have the following values: (-3, 1/8), (-2, 1/4), (-1, 1/2), (0, 1), (1, 2), (2, 4), and (3, 8).
$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x) = 2^{x}$	$\frac{1}{8}$	$\frac{1}{4}$	$\frac{1}{2}$	$1$	$2$	$4$	$8$

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 $f (x) = a b^{x},$ $b$ 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 $f (x) = 2^{x} .$

Graph of the exponential function, 2^(x), with labeled points at (-3, 1/8), (-2, ¼), (-1, ½), (0, 1), (1, 2), (2, 4), and (3, 8). The graph notes that the x-axis is an asymptote. — 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 $(0, \infty),$ 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 $g (x) = {(\frac{1}{2})}^{x} .$ Observe how the output values in Table 2 change as the input increases by $1.$

Table 2 Two rows and eight columns. The first row is labeled, “f(x)=2^x”, with the following values: (-3, 1/8), (-2, 1/4), (-1, 1/2), (0, 1), (1, 2), (2, 4), and (3, 8). The second row is labeled, “g(x)=log_2(x)”, with the following values: (1/8, -3), (1/4, -2), (1/2, -1), (1, 0), (2, 1), (4, 2), and (8, 3).
$x$	$-3$	$-2$	$-1$	$0$	$1$	$2$	$3$
$g (x) = (\frac{1}{2})^{x}$	$8$	$4$	$2$	$1$	$\frac{1}{2}$	$\frac{1}{4}$	$\frac{1}{8}$

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, $g (x) = {(\frac{1}{2})}^{x} .$

Graph of decreasing exponential function, (1/2)^x, with labeled points at (-3, 8), (-2, 4), (-1, 2), (0, 1), (1, 1/2), (2, 1/4), and (3, 1/8). The graph notes that the x-axis is an asymptote. — Figure 2

The domain of $g (x) = {(\frac{1}{2})}^{x}$ is all real numbers, the range is $(0, \infty),$ and the horizontal asymptote is $y = 0.$

Example 3

Sketching the Graph of an Exponential Function of the Form f(x) = b^x

Sketch a graph of $f (x) = {0.25}^{x} .$ State the domain, range, and asymptote.

Solution

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.

Table 3 Two rows and eight columns. The first row is labeled, “x”, and the second row is labeled, “f(x)=(0.25)^x”. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, Two rows and eight columns. The first row is labeled, “x”, and the second row is labeled, “f(x)=(0.25)^x”. Reading the columns as ordered pairs, we have the following values: (-3, 64), (-2, 16), (-1, 4), (0, 1), (1, 0.25), (2, 0.0625), and (3, 0.015625).
$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x) = {0.25}^{x}$	$64$	$16$	$4$	$1$	$0.25$	$0.0625$	$0.015625$

Plot the y-intercept, $(0, 1),$ along with two other points. We can use $(- 1, 4)$ and $(1, 0.25) .$

Draw a smooth curve connecting the points as in Figure 4.

Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25). — Figure 4

The domain is $(- \infty, \infty);$ the range is $(0, \infty);$ the horizontal asymptote is $y = 0.$

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.

Graph of three functions, g(x) = 2^x+3 in blue with an asymptote at y=3, f(x) = 2^x in orange with an asymptote at y=0, and h(x)=2^x-3 with an asymptote at y=-3. Note that each functions’ transformations are described in the text. — Figure 5

Observe the results of shifting $f (x) = 2^{x}$ vertically:

The domain, $(- \infty, \infty)$ remains unchanged.
When the function is shifted up 3 units to g(x)= 2 x +3:
- The y-intercept shifts up $3$ units to $(0, 4) .$
- The asymptote shifts up $3$ units to $y = 3.$
- The range becomes $(3, \infty) .$
When the function is shifted down 3 units to h(x)= 2 x −3:
- The y-intercept shifts down $3$ units to $(0, - 2) .$
- The asymptote also shifts down $3$ units to $y = - 3.$
- The range becomes $(- 3, \infty) .$

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.

Graph of three functions, g(x) = 2^(x+3) in blue, f(x) = 2^x in orange, and h(x)=2^(x-3). Each functions’ asymptotes are at y=0Note that each functions’ transformations are described in the text. — Figure 6

Observe the results of shifting $f (x) = 2^{x}$ horizontally:

The domain, $(- \infty, \infty),$ 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 $(0, 8) .$ 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 $(0, \frac{1}{8}) .$ Again, see that $2^{x - 3} = (\frac{1}{8}) 2^{x},$ so the initial value of the function is $\frac{1}{8} .$

Example 4

Graphing a Shift of an Exponential Function

Graph $f (x) = 2^{x + 1} - 3.$ State the domain, range, and asymptote.

Solution

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 $(- c, d),$ so the shift is $(- 1, −3) .$

Shift the graph of $f (x) = b^{x}$ left 1 units and down 3 units.

Graph of the function, f(x) = 2^(x+1)-3, with an asymptote at y=-3. Labeled points in the graph are (-1, -2), (0, -1), and (1, 1). — Figure 7

The domain is $(- \infty, \infty);$ the range is $(- 3, \infty);$ the horizontal asymptote is $y = −3.$

Example 5

Approximating the Solution of an Exponential Equation

Solve $42 = 1.2 {(5)}^{x} + 2.8$ graphically. Round to the nearest thousandth.

Solution

Press [Y=] and enter $1.2 {(5)}^{x} + 2.8$ next to Y₁=. Then enter 42 next to Y2=. For a window, use the values –3 to 3 for $x$ and –5 to 55 for $y .$ Press [GRAPH]. The graphs should intersect somewhere near $x = 2.$

For a better approximation, press [2ND] then [CALC]. Select [5: intersect] and press [ENTER] three times. The x-coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess?) To the nearest thousandth, $x \approx 2.166.$

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 $| a | > 0.$ 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.

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression. — 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 $\frac{1}{3} .$

Example 6

Graphing the Stretch of an Exponential Function

Sketch a graph of $f (x) = 4 {(\frac{1}{2})}^{x} .$ State the domain, range, and asymptote.

Solution

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 $f (x) = {(\frac{1}{2})}^{x}$ will be stretched by a factor of $4.$

Create a table of points as shown in Table 4.

Table 4 Two rows and eight columns. The first row is labeled, “x”, and the second row is labeled, “f(x)=4(0.25)^x”. Reading the columns as ordered pairs, we have the following values: (-3, 32), (-2, 16), (-1, 8), (0, 4), (1, 2), (2, 1), and (3, 0.5).
$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$f (x) = 4 (\frac{1}{2})^{x}$	$32$	$16$	$8$	$4$	$2$	$1$	$0.5$

Plot the y-intercept, $(0, 4),$ along with two other points. We can use $(- 1, 8)$ and $(1, 2) .$

Draw a smooth curve connecting the points, as shown in Figure 9.

Graph of the function, f(x) = 4(1/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2). — Figure 9

The domain is $(- \infty, \infty);$ the range is $(0, \infty);$ 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.

Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis. — 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.

Example 7

Writing and Graphing the Reflection of an Exponential Function

Find and graph the equation for a function, $g (x),$ that reflects $f (x) = {(\frac{1}{4})}^{x}$ about the x-axis. State its domain, range, and asymptote.

Solution

Since we want to reflect the parent function $f (x) = {(\frac{1}{4})}^{x}$ about the x-axis, we multiply $f (x)$ by $- 1$ to get, $g (x) = - {(\frac{1}{4})}^{x} .$ Next we create a table of points as in Table 5.

Table 5 Two rows and eight columns. The first row is labeled, “x”, and the second row is labeled, “f(x)=-(1/4)^x”. Reading the columns as ordered pairs, we have the following values: (-3, -64), (-2, -16), (-1, -4), (0, -1), (1, -0.25), (2, -0.0625), and (3, -0.0156).
$x$	$- 3$	$- 2$	$- 1$	$0$	$1$	$2$	$3$
$g (x) = - (\frac{1}{4})^{x}$	$- 64$	$- 16$	$- 4$	$- 1$	$- 0.25$	$- 0.0625$	$- 0.0156$

Plot the y-intercept, $(0, −1),$ along with two other points. We can use $(−1, −4)$ and $(1, −0.25) .$

Draw a smooth curve connecting the points:

Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25). — Figure 11

The domain is $(- \infty, \infty);$ the range is $(- \infty, 0);$ 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.

Table 6 Two rows and two columns. The first column shows the left shift of the equation g(x)=log_b(x) when b>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept remains (1, 0), the key point changes to (b^(-1), 1), the domain remains (0, infinity), and the range remains (-infinity, infinity). The second column shows the left shift of the equation g(x)=log_b(x) when b>1, and notes the following changes: the reflected function is decreasing as x moves from 0 to infinity, the asymptote remains x=0, the x-intercept changes to (-1, 0), the key point changes to (-b, 1), the domain changes to (-infinity, 0), and the range remains (-infinity, infinity).
Transformations of the Parent Function $f (x) = b^{x}$
Transformation	Form
Shift Horizontally $c$ units to the left Vertically $d$ units up	$f (x) = b^{x + c} + d$
Stretch and Compress Stretch if $\| a \| > 1$ Compression if $0 < \| a \| < 1$	$f (x) = a b^{x}$
Reflect about the x-axis	$f (x) = - b^{x}$
Reflect about the y-axis	$f (x) = b^{- x} = {(\frac{1}{b})}^{x}$
General equation for all transformations	$f (x) = a b^{x + c} + d$

Example 8

Writing a Function from a Description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

$f (x) = e^{x}$ is vertically stretched by a factor of $2$ , reflected across the y-axis, and then shifted up $4$ units.

Solution

We want to find an equation of the general form $f (x) = a b^{x + c} + d .$ We use the description provided to find $a,$ $b,$ $c,$ and $d .$

We are given the parent function $f (x) = e^{x},$ so $b = e .$
The function is stretched by a factor of $2$ , so $a = 2.$
The function is reflected about the y-axis. We replace $x$ with $- x$ to get: $e^{- x} .$
The graph is shifted vertically 4 units, so $d = 4.$

Substituting in the general form we get,

\begin{array}{l} f (x) & = a b^{x + c} + d \\ = 2 e^{- x + 0} + 4 \\ = 2 e^{- x} + 4 \end{array}

The domain is $(- \infty, \infty);$ the range is $(4, \infty);$ the horizontal asymptote is $y = 4.$

Key Equations

...
General Form for the Translation of the Parent Function $f (x) = b^{x}$	$f (x) = a b^{x + c} + d$

Key Concepts

The graph of the function $f (x) = b^{x}$ has a y-intercept at $(0, 1),$ domain $(- \infty, \infty),$ range $(0, \infty),$ and horizontal asymptote $y = 0.$ See Example 3.
If $b > 1,$ the function is increasing. The left tail of the graph will approach the asymptote $y = 0,$ and the right tail will increase without bound.
If $0 < b < 1,$ the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote $y = 0.$
The equation $f (x) = b^{x} + d$ represents a vertical shift of the parent function $f (x) = b^{x} .$
The equation $f (x) = b^{x + c}$ represents a horizontal shift of the parent function $f (x) = b^{x} .$ See Example 4.
Approximate solutions of the equation $f (x) = b^{x + c} + d$ can be found using a graphing calculator. See Example 5.
The equation $f (x) = a b^{x},$ where $a > 0,$ represents a vertical stretch if $| a | > 1$ or compression if $0 < | a | < 1$ of the parent function $f (x) = b^{x} .$ See Example 6.
When the parent function $f (x) = b^{x}$ is multiplied by $- 1,$ the result, $f (x) = - b^{x},$ is a reflection about the x-axis. When the input is multiplied by $- 1,$ the result, $f (x) = b^{- x},$ is a reflection about the y-axis. See Example 7.
All translations of the exponential function can be summarized by the general equation $f (x) = a b^{x + c} + d .$ See Table 3.
Using the general equation $f (x) = a b^{x + c} + d,$ we can write the equation of a function given its description. See Example 8.

Section Exercises

Verbal

Exercise 1

What role does the horizontal asymptote of an exponential function play in telling us about the end behavior of the graph?

Solution

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.

Exercise 2

What is the advantage of knowing how to recognize transformations of the graph of a parent function algebraically?

Algebraic

Exercise 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

$g (x) = 4 {(3)}^{- x};$ y-intercept: $(0, 4);$ Domain: all real numbers; Range: all real numbers greater than $0.$

Exercise 4

The graph of $f (x) = {(\frac{1}{2})}^{- x}$ 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.

Exercise 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

$g (x) = - 10^{x} + 7;$ y-intercept: $(0, 6);$ Domain: all real numbers; Range: all real numbers less than $7.$

Exercise 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.

Exercise 7

The graph of $f (x) = - \frac{1}{2} {(\frac{1}{4})}^{x - 2} + 4$ 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

$g (x) = 2 {(\frac{1}{4})}^{x};$ y-intercept: $(0, 2);$ 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.

Exercise 8

$f (x) = 3 {(\frac{1}{2})}^{x}$

Exercise 9

$g (x) = - 2 {(0.25)}^{x}$

Solution

y-intercept: $(0, - 2)$

Exercise 10

$h (x) = 6 {(1.75)}^{- x}$

For the following exercises, graph each set of functions on the same axes.

Exercise 11

$f (x) = 3 {(\frac{1}{4})}^{x},$ $g (x) = 3 {(2)}^{x},$ and $h (x) = 3 {(4)}^{x}$

Solution

Graph of three functions, g(x)=3(2)^(x) in blue, h(x)=3(4)^(x) in green, and f(x)=3(1/4)^(x) in orange.

Exercise 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.

Graph of six exponential functions. — Figure 12

Exercise 13

$f (x) = 2 {(0.69)}^{x}$

Solution

B

Exercise 14

$f (x) = 2 {(1.28)}^{x}$

Exercise 15

$f (x) = 2 {(0.81)}^{x}$

Solution

A

Exercise 16

$f (x) = 4 {(1.28)}^{x}$

Exercise 17

$f (x) = 2 {(1.59)}^{x}$

Solution

E

Exercise 18

$f (x) = 4 {(0.69)}^{x}$

For the following exercises, use the graphs shown in Figure 13. All have the form $f (x) = a b^{x} .$

Exercise 19

Which graph has the largest value for $b ?$

Solution

D

Exercise 20

Which graph has the smallest value for $b ?$

Exercise 21

Which graph has the largest value for $a ?$

Solution

C

Exercise 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.

Exercise 23

$f (x) = \frac{1}{2} {(4)}^{x}$

Solution

Graph of two functions, f(x)=(1/2)(4)^(x) in blue and -f(x)=(-1/2)(4)^x in orange.

Exercise 24

$f (x) = 3 {(0.75)}^{x} - 1$

Exercise 25

$f (x) = - 4 {(2)}^{x} + 2$

Solution

For the following exercises, graph the transformation of $f (x) = 2^{x} .$ Give the horizontal asymptote, the domain, and the range.

Exercise 26

$f (x) = 2^{- x}$

Exercise 27

$h (x) = 2^{x} + 3$

Solution

Horizontal asymptote: $h (x) = 3;$ Domain: all real numbers; Range: all real numbers strictly greater than $3.$

Exercise 28

$f (x) = 2^{x - 2}$

For the following exercises, describe the end behavior of the graphs of the functions.

Exercise 29

$f (x) = - 5 {(4)}^{x} - 1$

Solution

As $x \to \infty$ , $f (x) \to - \infty$ ;
As $x \to - \infty$ , $f (x) \to - 1$

Exercise 30

$f (x) = 3 {(\frac{1}{2})}^{x} - 2$

Exercise 31

$f (x) = 3 {(4)}^{- x} + 2$

Solution

As $x \to \infty$ , $f (x) \to 2$ ;
As $x \to - \infty$ , $f (x) \to \infty$

For the following exercises, start with the graph of $f (x) = 4^{x} .$ Then write a function that results from the given transformation.

Exercise 32

Shift $f (x)$ 4 units upward

Exercise 33

Shift $f (x)$ 3 units downward

Solution

$f (x) = 4^{x} - 3$

Exercise 34

Shift $f (x)$ 2 units left

Exercise 35

Shift $f (x)$ 5 units right

Solution

$f (x) = 4^{x - 5}$

Exercise 36

Reflect $f (x)$ about the x-axis

Exercise 37

Reflect $f (x)$ about the y-axis

Solution

$f (x) = 4^{- x}$

For the following exercises, each graph is a transformation of $y = 2^{x} .$ Write an equation describing the transformation.

Exercise 38

Graph of f(x)=2^(x) with the following translations: vertical stretch of 4, a reflection about the x-axis, and a shift up by 1.

Exercise 39

Graph of f(x)=2^(x) with the following translations: a reflection about the x-axis, and a shift up by 3.

Solution

$y = - 2^{x} + 3$

Exercise 40

Graph of f(x)=2^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis and y-axis, and a shift up by 3.

For the following exercises, find an exponential equation for the graph.

Exercise 41

Graph of f(x)=3^(x) with the following translations: vertical stretch of 2, a reflection about the x-axis, and a shift up by 7.

Solution

$y = - 2 {(3)}^{x} + 7$

Exercise 42

Graph of f(x)=(1/2)^(x) with the following translations: vertical stretch of 2, and a shift down by 4.

Numeric

For the following exercises, evaluate the exponential functions for the indicated value of $x .$

Exercise 43

$g (x) = \frac{1}{3} {(7)}^{x - 2}$ for $g (6) .$

Solution

$g (6) = 800 + \frac{1}{3} \approx 800.3333$

Exercise 44

$f (x) = 4 {(2)}^{x - 1} - 2$ for $f (5) .$

Exercise 45

$h (x) = - \frac{1}{2} {(\frac{1}{2})}^{x} + 6$ for $h (- 7) .$

Solution

$h (- 7) = - 58$

Technology

For the following exercises, use a graphing calculator to approximate the solutions of the equation. Round to the nearest thousandth.

Exercise 46

$- 50 = - {(\frac{1}{2})}^{- x}$

Exercise 47

$116 = \frac{1}{4} {(\frac{1}{8})}^{x}$

Solution

$x \approx - 2.953$

Exercise 48

$12 = 2 {(3)}^{x} + 1$

Exercise 49

$5 = 3 {(\frac{1}{2})}^{x - 1} - 2$

Solution

$x \approx - 0.222$

Exercise 50

$- 30 = - 4 {(2)}^{x + 2} + 2$

Extensions

Exercise 51

Explore and discuss the graphs of $F (x) = {(b)}^{x}$ and $G (x) = {(\frac{1}{b})}^{x} .$ Then make a conjecture about the relationship between the graphs of the functions $b^{x}$ and ${(\frac{1}{b})}^{x}$ for any real number $b > 0.$

Solution

The graph of $G (x) = {(\frac{1}{b})}^{x}$ 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 ${(\frac{1}{b})}^{x}$ is the the reflection about the y-axis, $F (- x) .$

Exercise 52

Prove the conjecture made in the previous exercise.

Exercise 53

Explore and discuss the graphs of $f (x) = 4^{x},$ $g (x) = 4^{x - 2},$ and $h (x) = (\frac{1}{16}) 4^{x} .$ Then make a conjecture about the relationship between the graphs of the functions $b^{x}$ and $(\frac{1}{b^{n}}) b^{x}$ for any real number n and real number $b > 0.$

Solution

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 $(\frac{1}{b^{n}}) b^{x}$ is the horizontal shift $f (x - n) .$

Exercise 54

Prove the conjecture made in the previous exercise.

Graphs of Exponential Functions

Learning Objectives

Objective 1: Graph exponential functions (IA 10.2.1).

Practice Makes Perfect

Objective 2: Function transformations (exponential). (CA 3.5.1-3.5.5)

Vertical and Horizontal Shifts:

Practice Makes Perfect

Graphing Exponential Functions

Sketching the Graph of an Exponential Function of the Form f(x) = bx

Graphing Transformations of Exponential Functions

Graphing a Vertical Shift

Graphing a Horizontal Shift

Graphing a Shift of an Exponential Function

Approximating the Solution of an Exponential Equation

Graphing a Stretch or Compression

Graphing the Stretch of an Exponential Function

Graphing Reflections

Writing and Graphing the Reflection of an Exponential Function

Summarizing Translations of the Exponential Function

Writing a Function from a Description

Key Equations

Key Concepts

Section Exercises

Verbal

Algebraic

Graphical

Numeric

Technology

Extensions

Sketching the Graph of an Exponential Function of the Form f(x) = b^x