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Mathematics LibreTexts

6.3 Graphs of Exponential Functions

Skills to Develop

  • Graph exponential functions.
  • Graph exponential functions using transformations.

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.

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 6.3.1 change as the input increases by \(1\).

\(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
\(f(x)=2^x\) \(\dfrac{1}{8}\) \(\dfrac{1}{4}\) \(\dfrac{1}{2}\) \(1\) \(2\) \(4\) \(8\)

Table 6.3.1

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 6.3.1 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 6.3.1 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 functiong \(g(x)={(\dfrac{1}{2})}^x\). Observe how the output values in Table 6.3.2 change as the input increases by \(1\).

\(x\) \(-3\) \(-2\) \(-1\) \(0\) \(1\) \(2\) \(3\)
\(g(x)={(\dfrac{1}{2})}^x\) \(8\) \(4\) \(2\) \(1\) \(\dfrac{1}{2}\) \(\dfrac{1}{4}\) \(\dfrac{1}{8}\)

Table 6.3.2

Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio \(\dfrac{1}{2}\).

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 6.3.2 shows the exponential decay function, \(g(x)={(\dfrac{1}{2})}^x\).

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 6.3.2

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

A General Note: CHARACTERISTICS OF THE GRAPH OF THE PARENT FUNCTION F(X) = BX

An exponential function with the form \(f(x)=b^x\), \(b>0\), \(b≠1\),has these characteristics:

  • one-to-one function
  • horizontal asymptote: \(y=0\)
  • domain: \((–\infty, \infty)\)
  • range: \((0,\infty)\)
  • x-intercept: none
  • y-intercept: \((0,1)\) 
  • increasing if \(b>1\)
  • decreasing if \(b<1\)

Figure 6.3.3 compares the graphs of exponential growth and decay functions.

Graph of two functions where the first graph is of a function of f(x) = b^x when b>1 and the second graph is of the same function when b is 0<b<1. Both graphs have the points (0, 1) and (1, b) labeled.

Figure 6.3.3

How to: Given an exponential function of the form \(f(x)=b^x\),graph the function.

  1. Create a table of points.
  2. Plot at least \(3\) point from the table, including the y-intercept \((0,1)\).
  3. Draw a smooth curve through the points.
  4. State the domain, \((−\infty,\infty)\),the range, \((0,\infty)\),and the horizontal asymptote,  \(y=0\).

Example

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.

 

\(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)
\(f(x)={0.25}^x\) \(64\) \(16\) \(4\) \(1\) \(0.25\) \(0.0625\0 \(0.015625\)

Table 6.3.3

  • 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 6.3.3.
  • 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 6.3.4.

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

Figure 6.3.4

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

Exercise 

Sketch the graph of \(f(x)=4^x\). State the domain, range, and asymptote.

Solution:

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

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

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 shiftd 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 6.3.5.

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 6.3.5

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, \(h(x)=2^{x−3}\). \(h(x)=2^{x−3}\). Both horizontal shifts are shown in Figure 6.3.6.

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 6.3.6

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 \((0,\dfrac{1}{8})\). Again, see that \(2^{x−3}=(\dfrac{1}{8})2^x\),so the initial value of the function is \(\dfrac{1}{8}\).

A General Note: SHIFTS OF THE PARENT FUNCTION F(X) = BX

For any constants \(c\) and \(d\),the function \(f(x)=b^{x+c}+d\) shifts the parent function \(f(x)=b^x\)

  • vertically \(d\) units, in the same direction of the sign of \(d\).
  • horizontally \(c\) units, in the opposite direction of the sign of \(c\).
  • The y-intercept becomes \((0,b^c+d)\).
  • The horizontal asymptote becomes \(y=d\).
  • The range becomes \((d,\infty)\).
  • The domain, \((−\infty,\infty)\),remains unchanged.

How to: Given an exponential function with the form \(f(x)=b^{x+c}+d\), graph the translation.

  1. Draw the horizontal asymptote \(y=d\).
  2. Identify the shift as \((−c,d)\). Shift the graph of \(f(x)=b^x\) left \(c\) units if \(c\) is positive, and right \(c\) units if \(c\) is negative.
  3. Shift the graph of \(f(x)=b^x\) up \(d\) units if \(d\) is positive, and down \(d\) units if \(d\) is negative.
  4. State the domain, \((−\infty,\infty)\),the range, \((d,\infty)\),and the horizontal asymptote \(y=d\).

Example

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 6.3.7

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

Exercise 

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

Solution:

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

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).

How to: Given an equation of the form \(f(x)=b^{x+c}+d\) for \(x\), use a graphing calculator to approximate the solution.

  • Press [Y=]. Enter the given exponential equation in the line headed “Y1=”.
  • Enter the given value forf(x) f(x) in the line headed “Y2=”.
  • Press [WINDOW]. Adjust the y-axis so that it includes the value entered for “Y2=”.
  • Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value off(x). f(x).
  • To find the value ofx, x,we compute the point of intersection. Press [2ND] then [CALC]. Select “intersect” and press [ENTER] three times. The point of intersection gives the value of for the indicated value of the function.

Example

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 nearx=2. 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, \(x≈2.166\).

Exercise 

Solve \(4=7.85{(1.15)}^x−2.27\) graphically. Round to the nearest thousandth.

Solution:

\(x≈−1.608\)

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 6.3.8, and the compression, using \(a=\dfrac{1}{3}\),to get \(h(x)=\dfrac{1}{3}{(2)}^x\) as shown on the right in Figure 6.3.8.

Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.

Figure 6.3.8 (a) \(g(x)=3{(2)}^x\) stretches the graph of \(f(x)=2^x\) vertically by a factor of \(3\). (b) \(h(x)=\dfrac{1}{3}{(2)}^x\) compresses the graph of \(f(x)=2^x\) vertically by a factor of \(\dfrac{1}{3}\).

A General Note: STRETCHES AND COMPRESSIONS OF THE PARENT FUNCTION F(X) = BX

For any factor \(a>0\),the function \(f(x)=a{(b)}^x\)

  • is stretched vertically by a factor of \(a\) if \(|a|>1\).
  • is compressed vertically by a factor of \(a\) if \(|a|<1\).
  • has a y-intercept of \((0,a)\).
  • has a horizontal asymptote at \(y=0\), a range of \((0,\infty)\), and a domain of \((−\infty,\infty)\),which are unchanged from the parent function.

Example

Graphing the Stretch of an Exponential Function

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

Solution:

Before graphing, identify the behavior and key points on the graph.

Table 6.3.4

  • Since \(b=\dfrac{1}{2}\) 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 \(f(x)={(\dfrac{1}{2})}^x\) will be stretched by a factor of \(4\).
  • Create a table of points as shown in Table 6.3.4.
    \(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)

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

    \(32\) \(16\) \(8\) \(4\) \(2\) \(1\) \(0.5\)
  • 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 6.3.9.

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 6.3.9

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

Exercise 

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

Solution:

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

Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).

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 6.3.10, and the reflection about the y-axis \(h(x)=2^{−x}\), is shown on the right side of Figure 6.3.10.

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 6.3.10 (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.

A General Note: REFLECTIONS OF THE PARENT FUNCTION F(X) = BX

The function \(f(x)=−b^x\)

  • reflects the parent function \(f(x)=b^x\) about the x-axis.
  • has a y-intercept of \((0,−1)\).
  • has a range of \((−\infty,0)\)
  • has a horizontal asymptote at \(y=0\) and domain of \((−\infty,\infty)\),which are unchanged from the parent function.

The function \(f(x)=b^{−x}\)

  • reflects the parent function \(f(x)=b^x\) about the y-axis.
  • has a y-intercept of \((0,1)\), a horizontal asymptote at \(y=0\), a range of \((0,\infty)\), and a domain of \((−\infty,\infty)\), which are unchanged from the parent function.

Example

Writing and Graphing the Reflection of an Exponential Function

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

Solution:

Since we want to reflect the parent function \(f(x)={(\dfrac{1}{4})}^x\) about the x-axis, we multiply \(f(x)\) by \(−1\) to get, \(g(x)=−{(\dfrac{1}{4})}^x\). Next we create a table of points as in Table 6.3.5.

\(x\) \(−3\) \(−2\) \(−1\) \(0\) \(1\) \(2\) \(3\)

\(g(x)=−{(\dfrac{1}{4})}^x\)

\(−64\)

\(−16\)

\(−4\)

\(−1\)

\(−0.25\)

\(−0.0625\)

\(−0.0156\)

Table 6.3.5

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 6.3.11

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

Exercise 

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:

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

Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).

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 6.3.6 to arrive at the general equation for translating exponential functions.

Translations of the Parent Function \(f(x)=b^x\)
Translation 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 \(0<|a|<1\)

\(f(x)=ab^x\)

Reflect about the x-axis

\(f(x)=−b^x\)

Reflect about the y-axis

\(f(x)=b^{−x}={(\dfrac{1}{b})}^x\)

General equation for all translations

\(f(x)=ab^{x+c}+d\)

Table 6.3.6

A General Note: TRANSLATIONS OF EXPONENTIAL FUNCTIONS

A translation of an exponential function has the form

\(f(x)=ab^{x+c}+d\)

Where the parent function, \(y=b^x\), \(b>1\),is

  • shifted horizontally \(c\) units to the left.
  • stretched vertically by a factor of \(|a|\) if \(|a|>0\). 
  • compressed vertically by a factor of \(|a|\) if \(0<|a|<1\).
  • shifted vertically \(d\) units.
  • reflected about the x-axis when \(a<0\).

Note the order of the shifts, transformations, and reflections follow the order of operations.

Example

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)=abx+c+d.  f(x)=abx+c+d. We use the description provided to finda, a, b,b, c,c, and d. 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,

\(f(x)=ab^{x+c}+d\)

\(=2e^{−x+0}+4\)

\(=2e^{−x}+4\)

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

Exercise 

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

\(f(x)=e^x\) is compressed vertically by a factor of \(\dfrac{1}{3}\), reflected across the x-axis and then shifted down \(2\) units.

Solution:

\(f(x)=−\dfrac{1}{3}e^{x}−2\); the domain is \((−\infty,\infty)\); the range is \((−\infty,2)\); the horizontal asymptote is \(y=2\).

Media

Access this online resource for additional instruction and practice with graphing exponential functions.

Key Equations

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

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.
  • 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.
  • Approximate solutions of the equation \(f(x)=b^{x+c}+d\) can be found using a graphing calculator. See Example.
  • The equation \(f(x)=ab^x\), 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.
  • 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.
  • All translations of the exponential function can be summarized by the general equation \(f(x)=ab^{x+c}+d\). See Table.
  • Using the general equation \(f(x)=ab^{x+c}+d\), we can write the equation of a function given its description. See Example.