4.2: Graphs of Exponential Functions
 Page ID
 18495
Skills to Develop
 Graph exponential functions.
 Graph exponential functions using transformations.
As we discussed in the previous section, exponential functions are used for many realworld applications such as finance, forensics, computer science, and most of the life sciences. Working with an equation that describes a realworld 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 \(\PageIndex{1}\) change as the input increases by \(1\).
\(x\)  \(−3\)  \(−2\)  \(−1\)  \(0\)  \(1\)  \(2\)  \(3\) 

\(f(x)=2^x\)  \(\dfrac{1}{8}\)  \(\dfrac{1}{4}\)  \(\dfrac{1}{2}\)  \(1\)  \(2\)  \(4\)  \(8\) 
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 \(\PageIndex{1}\) shows the exponential growth function \(f(x)=2^x\).
Figure \(\PageIndex{1}\): 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 \(y=0\). Note that, unlike a horizontal asymptote for a rational function (see Section 3.7), the graph approaches \(y=0\) only at one end.
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 \(g(x)={\left(\dfrac{1}{2}\right)}^x\). Observe how the output values in Table \(\PageIndex{2}\) change as the input increases by \(1\).
\(x\)  \(3\)  \(2\)  \(1\)  \(0\)  \(1\)  \(2\)  \(3\) 

\(g(x)={\left(\dfrac{1}{2}\right)}^x\)  \(8\)  \(4\)  \(2\)  \(1\)  \(\dfrac{1}{2}\)  \(\dfrac{1}{4}\)  \(\dfrac{1}{8}\) 
Again, because the input is increasing by \(1\), each output value is the product of the previous output and the base, or constant ratio \(\dfrac{1}{2}\).
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 \(\PageIndex{2}\) shows the exponential decay function, \(g(x)={\left(\dfrac{1}{2}\right)}^x\).
Figure \(\PageIndex{2}\)
The domain of \(g(x)=\left(\frac{1}{2}\right)^x\) is all real numbers, the range is \((0,\infty)\), and the horizontal asymptote is \(y=0\).
CHARACTERISTICS OF THE GRAPH OF THE toolkit FUNCTION \(f(x) = b^x\)
An exponential function with the form \(f(x)=b^x\), \(b>0\), \(b≠1\), has these characteristics:
 onetoone 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 \(\PageIndex{3}\) compares the graphs of exponential growth and decay functions.
Figure \(\PageIndex{3}\)
Given an exponential function of the form \(f(x)=b^x\), graph the function
 Plot at least \(3\) points of the graph by finding 3 inputoutput pairs, including the \(y\)intercept \((0,1)\).
 Draw a smooth curve through the points.
 State the domain, \((−\infty,\infty)\), the range, \((0,\infty)\), and the horizontal asymptote, \(y=0\).
Example \(\PageIndex{1}\): 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
Since \(b=0.25\) is between zero and one, we know the function is decreasing. The end behavior of the graph is as follows: as \(x \rightarrow \infty\), \(y \rightarrow \infty\), and as \(x \rightarrow \infty\), \(y \rightarrow 0\), so the graph has an asymptote \(y=0\). Find two other points.
 Plot the \(y\)intercept, \((0,1)\), along with the two other points. We will use \((−1,4)\) and \((1,0.25)\).
Draw a smooth curve connecting the points as in Figure \(\PageIndex{4}\).
Figure \(\PageIndex{4}\)
The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).
\(\PageIndex{1}\) Sketch the graph of \(f(x)=4^x\). State the domain, range, and asymptote.
 Answer

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 toolkit functions, we can apply the four types of transformations—shifts, reflections, stretches, and compressions—to the toolkit 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 toolkit 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 toolkit 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 \(\PageIndex{5}\).
Figure \(\PageIndex{5}\)
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 yintercept 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 yintercept 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 toolkit 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 toolkit 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 \(\PageIndex{6}\).
Figure \(\PageIndex{6}\)
Observe the results of shifting \(f(x)=2^x\) horizontally:
 The domain, \((−\infty,\infty)\), remains unchanged.
 The asymptote, \(y=0\), remains unchanged.
 The yintercept shifts such that:
 When the function is shifted left \(3\) units to \(g(x)=2^{x+3}\),the yintercept becomes \((0,8)\). This is because \(2^{x+3}=2^32^x=(8)2^x\), so the initial value of the function is \(8\).
 When the function is shifted right \(3\) units to \(h(x)=2^{x−3}\),the yintercept becomes \(\left(0,\dfrac{1}{8}\right)\). Again, see that \(2^{x−3}=\left(\dfrac{1}{8}\right)2^x\), so the initial value of the function is \(\dfrac{1}{8}\).
SHIFTS OF THE toolkit FUNCTION \(f(x) = b^x\)
For any constants \(c\) and \(d\),the function \(f(x)=b^{x+c}+d\) shifts the toolkit 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 yintercept becomes \((0,b^c+d)\).
 The horizontal asymptote becomes \(y=d\).
 The range becomes \((d,\infty)\).
 The domain, \((−\infty,\infty)\),remains unchanged.
Given an exponential function with the form \(f(x)=b^{x+c}+d\), graph the translation
 Draw the horizontal asymptote \(y=d\).
 Shift the graph of \(f(x)=b^x\) left \(c\) units if \(c\) is positive, and right \(c\) units if \(c\) is negative.
 Shift the graph of \(f(x)=b^x\) up \(d\) units if \(d\) is positive, and down \(d\) units if \(d\) is negative.
 State the domain, \((−\infty,\infty)\), the range, \((d,\infty)\), and the horizontal asymptote \(y=d\).
Example \(\PageIndex{2}\): 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 \(y=−3\).
Shift the graph of \(f(x)=b^x\) left \(1\) unit and down \(3\) units.Figure \(\PageIndex{7}\)
The domain is \((−\infty,\infty)\); the range is \((−3,\infty)\); the horizontal asymptote is \(y=−3\).
\(\PageIndex{2}\) Graph \(f(x)=2^{x−1}+3\). State domain, range, and asymptote.
 Answer

The domain is \((−\infty,\infty)\); the range is \((3,\infty)\); the horizontal asymptote is \(y=3\).
Given an equation of the form \(f(x)=b^{x+c}+d\), use a graphing calculator to approximate the solution for \(x\)
 Press [Y=]. Enter the given exponential equation in the line headed “Y_{1}=”.
 Enter the given value for \(f(x)\) in the line headed “Y_{2}=”.
 Press [WINDOW]. Adjust the \(y\)axis so that it includes the value entered for “Y_{2}=”.
 Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of \(f(x)\).
 To find the value of \(x\),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 \(x\)for the indicated value of the function.
Example \(\PageIndex{3}\): 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 Y_{1}=. 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 xcoordinate 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, \(x≈2.166\).
\(\PageIndex{3}\) Solve \(4=7.85{(1.15)}^x−2.27\) graphically. Round to the nearest thousandth.
 Answer

\(x≈−1.608\)
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 toolkit function \(f(x)=b^x\) by a constant \(a>0\). For example, if we begin by graphing the toolkit 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 \(\PageIndex{8}\), and the compression, using \(a=\frac{1}{3}\), to get \(h(x)=\frac{1}{3}{(2)}^x\) as shown on the right in Figure \(\PageIndex{8}\).
Figure \(\PageIndex{8}\): (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 \(\frac{1}{3}\).
STRETCHES AND COMPRESSIONS OF THE toolkit FUNCTION \(f(x)=b^x\)
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 yintercept 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 \(\PageIndex{4}\): 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
Before graphing, identify the behavior and key points on the graph.
 Since \(b=\frac{1}{2}\) is between zero and one, the graph will increase without bound as \(x \rightarrow \infty\), and the graph will approach the \(x\)axis as \(x \rightarrow \infty\).
 Since \(a=4\),the graph of \(f(x)=\left(\frac{1}{2}\right)^x\) will be stretched by a factor of \(4\).
 Plot the yintercept, \((0,4)\), along with two other points. We will use \((−1,8)\) and \((1,2)\).
Draw a smooth curve connecting the points, as shown in Figure \(\PageIndex{9}\).
Figure \(\PageIndex{9}\)
The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); the horizontal asymptote is \(y=0\).
\(\PageIndex{4}\) Sketch the graph of \(f(x)=\frac{1}{2}{(4)}^x\). State the domain, range, and asymptote.
 Answer

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 across the xaxis or the yaxis. When we multiply the toolkit function \(f(x)=b^x\) by \(−1\),we get a reflection across the xaxis. When we multiply the input by \(−1\),we get a reflection across the yaxis. For example, if we begin by graphing the toolkit function \(f(x)=2^x\), we can then graph the two reflections alongside it. The reflection across the \(x\)axis, \(g(x)=−2^x\), is shown on the left side of Figure \(\PageIndex{10}\), and the reflection across the \(y\)axis \(h(x)=2^{−x}\), is shown on the right side of Figure \(\PageIndex{10}\).
Figure \(\PageIndex{10}\): (a) \(g(x)=−2^x\) reflects the graph of \(f(x)=2^x\) across the xaxis. (b) \(g(x)=2^{−x}\) reflects the graph of \(f(x)=2^x\) across the \(y\)axis.
REFLECTIONS OF THE toolkit FUNCTION \(f(x) = b^x\)
The function \(f(x)=−b^x\)
 reflects the toolkit function \(f(x)=b^x\) across the xaxis.
 has a yintercept 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 toolkit function \(f(x)=b^x\) across 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 toolkit function.
Example \(\PageIndex{5}\): Writing and Graphing the Reflection of an Exponential Function
Find and graph the equation for a function \(g(x)\) that reflects \(f(x)={(\frac{1}{4})}^x\) across the xaxis. State its domain, range, and asymptote.
Solution
Since we want to reflect the parent function \(f(x)={(\dfrac{1}{4})}^x\) about the xaxis, we multiply \(f(x)\) by \(−1\) to get, \(g(x)=−{(\dfrac{1}{4})}^x\). Plot the yintercept, \((0,−1)\),along with two other points. We will use \((−1,−4)\) and \((1,−0.25)\).
Draw a smooth curve connecting the points:
Figure \(\PageIndex{11}\)
The domain is \((−\infty,\infty)\); the range is \((−\infty,0)\); the horizontal asymptote is \(y=0\).
\(\PageIndex{5}\)
Find and graph the equation for a function, \(g(x)\), that reflects \(f(x)={1.25}^x\) across the yaxis. State its domain, range, and asymptote.
 Answer

The domain is \((−\infty,\infty)\); the range is \((0,\infty)\); 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 \(\PageIndex{3}\) to arrive at the general equation for translating exponential functions.
TRANSLATIONS OF EXPONENTIAL FUNCTIONS
A translation of an exponential function has the form
\(f(x)=ab^{x+c}+d\)
Where the toolkit 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 across the xaxis when \(a<0\).
Note that the order of the shifts, transformations, and reflections follows the order of operations.
Example \(\PageIndex{6}\): 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 yaxis, and then shifted up \(4\) units.
Solution
We want to find an equation of the general form \(f(x)=ab^{x+c}+d\). We use the description provided to find \(a, b, c,\) and \(d\).
 We are given the toolkit function \(f(x)=e^x\), so \(b=e\). Note: \(e\) is a number, not a variable. It was defined in Section 4.1. Write down its definition, hand it it to your professor, and make her give you extra credit!
 The function is stretched by a factor of \(2\), so \(a=2\).
 The function is reflected about the yaxis. We replace \(x\) with \(−x\) to get: \(e^{−x}\).
 There is no horizontal shift, so \(c=0\).
 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 \(y=4\).
\(\PageIndex{6}\) 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 xaxis and then shifted down \(2\) units.
 Answer

\(f(x)=−\frac{1}{3}e^{x}−2\); 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 toolkit function \(f(x)=b^x\):  \(f(x)=ab^{x+c}+d\) 
Key Concepts
 The graph of the function \(f(x)=b^x\) has a yintercept at \((0, 1)\), domain \((−\infty, \infty)\), range \((0, \infty)\), and horizontal asymptote \(y=0\).
 If \(b>1\), the function is increasing. The end behavior of the graph to the left: \(y\) will approach the asymptote \(y=0\), and to the right: \(y\) will increase without bound.
 If \(0<b<1\), the function is decreasing. The end behavior of the graph to the left: \(y\)will increase without bound, and to the right: \(y\) will approach the asymptote \(y=0\).
 The equation \(f(x)=b^x+d\) represents a vertical shift of the toolkit function \(f(x)=b^x\).
 The equation \(f(x)=b^{x+c}\) represents a horizontal shift of the toolkit function \(f(x)=b^x\).
 Approximate solutions of the equation \(f(x)=b^{x+c}+d\) can be found using a graphing calculator.
 The equation \(f(x)=ab^x\), where \(a>0\), represents a vertical stretch if \(a>1\) or compression if \(0<a<1\) of the toolkit function \(f(x)=b^x\).
 When the toolkit function \(f(x)=b^x\) is multiplied by \(−1\), the result, \(f(x)=−b^x\), is a reflection across the xaxis. When the input is multiplied by \(−1\),the result, \(f(x)=b^{−x}\), is a reflection across the yaxis.
 All translations of the exponential function can be summarized by the general equation \(f(x)=ab^{x+c}+d\).
 Using the general equation \(f(x)=ab^{x+c}+d\), we can write the equation of a function given its description.
Contributors
 Lynn Marecek (Santa Ana College) and MaryAnne AnthonySmith (formerly of Santa Ana College). This content produced by OpenStax and is licensed under a Creative Commons Attribution License 4.0 license.