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Guides d'étude > College Algebra

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.

Characteristics of Graphs of 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 [latex]f\left(x\right)={b}^{x}[/latex] whose base is greater than one. We’ll use the function [latex]f\left(x\right)={2}^{x}[/latex]. Observe how the output values in the table below change as the input increases by 1.
x –3 –2 –1 0 1 2 3
[latex]f\left(x\right)={2}^{x}[/latex] [latex]\frac{1}{8}[/latex] [latex]\frac{1}{4}[/latex] [latex]\frac{1}{2}[/latex] 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 [latex]f\left(x\right)=a{b}^{x}[/latex], 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.
The graph below shows the exponential growth function [latex]f\left(x\right)={2}^{x}[/latex].
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. Notice that the graph gets close to the x-axis, but never touches it.
The domain of [latex]f\left(x\right)={2}^{x}[/latex] is all real numbers, the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex]. To get a sense of the behavior of exponential decay, we can create a table of values for a function of the form [latex]f\left(x\right)={b}^{x}[/latex] whose base is between zero and one. We’ll use the function [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex]. Observe how the output values in the table below change as the input increases by 1.
x –3 –2 –1 0 1 2 3
[latex]g\left(x\right)=\left(\frac{1}{2}\right)^{x}[/latex] 8 4 2 1 [latex]\frac{1}{2}[/latex] [latex]\frac{1}{4}[/latex] [latex]\frac{1}{8}[/latex]
Again, because the input is increasing by 1, each output value is the product of the previous output and the base, or constant ratio [latex]\frac{1}{2}[/latex]. 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.
The graph shows the exponential decay function, [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex].
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. The domain of [latex]g\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex] is all real numbers, the range is [latex]\left(0,\infty \right)[/latex], and the horizontal asymptote is [latex]y=0[/latex].

A General Note: Characteristics of the Graph of the Parent Function f(x) = bx

An exponential function with the form [latex]f\left(x\right)={b}^{x}[/latex], [latex]b>0[/latex], [latex]b\ne 1[/latex], has these characteristics:
  • one-to-one function
  • horizontal asymptote: [latex]y=0[/latex]
  • domain: [latex]\left(-\infty , \infty \right)[/latex]
  • range: [latex]\left(0,\infty \right)[/latex]
  • x-intercept: none
  • y-intercept: [latex]\left(0,1\right)[/latex]
  • increasing if [latex]b>1[/latex]
  • decreasing if [latex]b<1[/latex]
Use the sliders in the graph below to compare the graphs of exponential growth and decay functions. Which one is growth and which one is decay? https://www.desmos.com/calculator/3pqyjsunkg

How To: Given an exponential function of the form [latex]f\left(x\right)={b}^{x}[/latex], graph the function.

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

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

Sketch a graph of [latex]f\left(x\right)={0.25}^{x}[/latex]. State the domain, range, and asymptote.

Answer: 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.
    x –3 –2 –1 0 1 2 3
    [latex]f\left(x\right)={0.25}^{x}[/latex] 64 16 4 1 0.25 0.0625 0.015625
  • Plot the y-intercept, [latex]\left(0,1\right)[/latex], along with two other points. We can use [latex]\left(-1,4\right)[/latex] and [latex]\left(1,0.25\right)[/latex].
Draw a smooth curve connecting the points.
Graph of the decaying exponential function f(x) = 0.25^x with labeled points at (-1, 4), (0, 1), and (1, 0.25). The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].

Try It

Sketch the graph of [latex]f\left(x\right)={4}^{x}[/latex]. State the domain, range, and asymptote.

Answer: The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].

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 [latex]f\left(x\right)={b}^{x}[/latex] 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 [latex]f\left(x\right)={b}^{x}[/latex], giving us a vertical shift d units in the same direction as the sign. For example, if we begin by graphing a parent function, [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two vertical shifts alongside it, using [latex]d=3[/latex]: the upward shift, [latex]g\left(x\right)={2}^{x}+3[/latex] and the downward shift, [latex]h\left(x\right)={2}^{x}-3[/latex]. Both vertical shifts are shown in the figure below. 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. Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] vertically:
  • The domain, [latex]\left(-\infty ,\infty \right)[/latex] remains unchanged.
  • When the function is shifted up 3 units to [latex]g\left(x\right)={2}^{x}+3[/latex]:
    • The y-intercept shifts up 3 units to [latex]\left(0,4\right)[/latex].
    • The asymptote shifts up 3 units to [latex]y=3[/latex].
    • The range becomes [latex]\left(3,\infty \right)[/latex].
  • When the function is shifted down 3 units to [latex]h\left(x\right)={2}^{x}-3[/latex]:
    • The y-intercept shifts down 3 units to [latex]\left(0,-2\right)[/latex].
    • The asymptote also shifts down 3 units to [latex]y=-3[/latex].
    • The range becomes [latex]\left(-3,\infty \right)[/latex].

Try it

  1. Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 4 units up. Add a line that represents the horizontal asymptote for this function. What is the equation for this function? What is the new y-intercept? What is it's domain and range?
  2. Now create a graph of the function [latex]f(x) = 2^x[/latex] that has been shifted down 2 units. Add a line that represents the horizontal asymptote.What is the equation for this function? What is the new y-intercept? What is it's domain and range?
https://www.desmos.com/calculator/5mrjqegkxk

Answer:

  1. Equation: [latex]f(x) = 2^x+4[/latex], Horizontal Asymptote: [latex]y = 4[/latex], y-intercept: [latex](0,5)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](4,\infty)[/latex]
  2. Equation: [latex]f(x) = 2^x-2[/latex], Horizontal Asymptote: [latex]y = -2[/latex], y-intercept: [latex](0,-1)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](-2,\infty)[/latex]

 

Graphing a Horizontal Shift

The next transformation occurs when we add a constant c to the input of the parent function [latex]f\left(x\right)={b}^{x}[/latex], giving us a horizontal shift c units in the opposite direction of the sign. For example, if we begin by graphing the parent function [latex]f\left(x\right)={2}^{x}[/latex], we can then graph two horizontal shifts alongside it, using [latex]c=3[/latex]: the shift left, [latex]g\left(x\right)={2}^{x+3}[/latex], and the shift right, [latex]h\left(x\right)={2}^{x - 3}[/latex]. Both horizontal shifts are shown in the figure below. 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. Observe the results of shifting [latex]f\left(x\right)={2}^{x}[/latex] horizontally:
  • The domain, [latex]\left(-\infty ,\infty \right)[/latex], remains unchanged.
  • The asymptote, [latex]y=0[/latex], remains unchanged.
  • The y-intercept shifts such that:
    • When the function is shifted left 3 units to [latex]g\left(x\right)={2}^{x+3}[/latex], the y-intercept becomes [latex]\left(0,8\right)[/latex]. This is because [latex]{2}^{x+3}=\left(8\right){2}^{x}[/latex], so the initial value of the function is 8.
    • When the function is shifted right 3 units to [latex]h\left(x\right)={2}^{x - 3}[/latex], the y-intercept becomes [latex]\left(0,\frac{1}{8}\right)[/latex]. Again, see that [latex]{2}^{x - 3}=\left(\frac{1}{8}\right){2}^{x}[/latex], so the initial value of the function is [latex]\frac{1}{8}[/latex].

try it

  1. Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 4 units to the right. What is the equation for this function? What is the new y-intercept? What are it's domain and range?
  2. Use the slider in the graph below to create a graph of [latex]f(x) = 2^x[/latex] that has been shifted 3 units to the left. What is the equation for this function? What is the new y-intercept? What are it's domain and range?
https://www.desmos.com/calculator/rpv1kea0pz

Answer:

  1. Equation: [latex]f(x) = 2^{x-4}[/latex], y-intercept: [latex](0,\frac{1}{32})[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](0,\infty)[/latex]
  2. Equation: [latex]f(x) = 2^{x+3}[/latex], y-intercept: [latex](0,8)[/latex]Domain: [latex](-\infty,\infty)[/latex], Range: [latex](0,\infty)[/latex]

 

A General Note: Shifts of the Parent Function [latex]f\left(x\right)={b}^{x}[/latex]

For any constants c and d, the function [latex]f\left(x\right)={b}^{x+c}+d[/latex] shifts the parent function [latex]f\left(x\right)={b}^{x}[/latex]
  • 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 [latex]\left(0,{b}^{c}+d\right)[/latex].
  • The horizontal asymptote becomes y = d.
  • The range becomes [latex]\left(d,\infty \right)[/latex].
  • The domain, [latex]\left(-\infty ,\infty \right)[/latex], remains unchanged.

How To: Given an exponential function with the form [latex]f\left(x\right)={b}^{x+c}+d[/latex], graph the translation.

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

Example: Graphing a Shift of an Exponential Function

Graph [latex]f\left(x\right)={2}^{x+1}-3[/latex]. State the domain, range, and asymptote.

Answer: We have an exponential equation of the form [latex]f\left(x\right)={b}^{x+c}+d[/latex], with [latex]b=2[/latex], [latex]c=1[/latex], and [latex]d=-3[/latex]. Draw the horizontal asymptote [latex]y=d[/latex], so draw [latex]y=-3[/latex]. Identify the shift as [latex]\left(-c,d\right)[/latex], so the shift is [latex]\left(-1,-3\right)[/latex].

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). The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-3,\infty \right)[/latex]; the horizontal asymptote is [latex]y=-3[/latex].
Shift the graph of [latex]f\left(x\right)={b}^{x}[/latex] left 1 units and down 3 units. The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-3,\infty \right)[/latex]; the horizontal asymptote is [latex]y=-3[/latex].

Try It

Use the sliders in the graph below to create a graph of the function [latex]f\left(x\right)={2}^{x - 1}+3[/latex]. State domain, range, and asymptote. https://www.desmos.com/calculator/e5l4eca3ob

Answer: The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(3,\infty \right)[/latex]; the horizontal asymptote is y = 3.

In the following video, we show more examples of the difference between horizontal and vertical translations of exponential functions and the resultant graphs and equations. https://youtu.be/phYxEeJ7ZW4

Use a Graph to Approximate a Solution to an Exponential Equation

Graphing can help you confirm or find the solution to an exponential equation. An exponential equation is different from a function because a function is a large collection of points made of inputs and corresponding outputs, whereas equations that you have seen typically have one, two, or no solutions.  For example, [latex]f(x)=2^{x}[/latex] is a function and is comprised of many points [latex](x,f(x))[/latex], and [latex]4=2^{x}[/latex] can be solved to find the specific value for x that makes it a true statement. The graph below shows the intersection of the line [latex]f(x)=4[/latex], and [latex]f(x)=2^{x}[/latex], you can see they cross at [latex]y=4[/latex]. https://www.desmos.com/calculator/lhmpdkbjt0   In the next example, you can try this for yourself.

Example : Approximating the Solution of an Exponential Equation

Use Desmos to solve [latex]42=1.2{\left(5\right)}^{x}+2.8[/latex] graphically.

Answer: First, graph the function [latex]f(x)=1.2{\left(5\right)}^{x}+2.8[/latex] , then add another function [latex]f(x) = 42[/latex]. Desmos automatically calculates points of interest  including intersections.  Essentially, you are looking for the intersection of two functions. Click on the point of intersection, and you will see the the x and y values for the point. Your graph will look like this: https://www.desmos.com/calculator/ozaejvejqn

Try It

Solve [latex]4=7.85{\left(1.15\right)}^{x}-2.27[/latex] graphically. Round to the nearest thousandth.

Answer: [latex]x\approx -1.608[/latex]

 

Stretch, Compress, or Reflect an Exponential Function

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 [latex]f\left(x\right)={b}^{x}[/latex] by a constant [latex]|a|>0[/latex]. For example, if we begin by graphing the parent function [latex]f\left(x\right)={2}^{x}[/latex], we can then graph the stretch, using [latex]a=3[/latex], to get [latex]g\left(x\right)=3{\left(2\right)}^{x}[/latex] as shown on the left in Figure 8, and the compression, using [latex]a=\frac{1}{3}[/latex], to get [latex]h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}[/latex] as shown on the right in the figure below.
Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression. (a) [latex]g\left(x\right)=3{\left(2\right)}^{x}[/latex] stretches the graph of [latex]f\left(x\right)={2}^{x}[/latex] vertically by a factor of 3. (b) [latex]h\left(x\right)=\frac{1}{3}{\left(2\right)}^{x}[/latex] compresses the graph of [latex]f\left(x\right)={2}^{x}[/latex] vertically by a factor of [latex]\frac{1}{3}[/latex].

A General Note: Stretches and Compressions of the Parent Function f(x) = bx

For any factor a > 0, the function [latex]f\left(x\right)=a{\left(b\right)}^{x}[/latex]
  • is stretched vertically by a factor of a if [latex]|a|>1[/latex].
  • is compressed vertically by a factor of a if [latex]|a|<1[/latex].
  • has a y-intercept of [latex]\left(0,a\right)[/latex].
  • has a horizontal asymptote at [latex]y=0[/latex], a range of [latex]\left(0,\infty \right)[/latex], and a domain of [latex]\left(-\infty ,\infty \right)[/latex], which are unchanged from the parent function.

Example: Graphing the Stretch of an Exponential Function

Sketch a graph of [latex]f\left(x\right)=4{\left(\frac{1}{2}\right)}^{x}[/latex]. State the domain, range, and asymptote.

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

  • Since [latex]b=\frac{1}{2}[/latex] 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 [latex]f\left(x\right)={\left(\frac{1}{2}\right)}^{x}[/latex] will be stretched by a factor of 4.
  • Create a table of points.
    x –3 –2 –1 0 1 2 3
    [latex]f\left(x\right)=4\left(\frac{1}{2}\right)^{x}[/latex] 32 16 8 4 2 1 0.5
  • Plot the y-intercept, [latex]\left(0,4\right)[/latex], along with two other points. We can use [latex]\left(-1,8\right)[/latex] and [latex]\left(1,2\right)[/latex].
  • Draw a smooth curve connecting the points.
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). The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is y = 0.

Try It

Use the sliders in the graph below to sketch the graph of [latex]f\left(x\right)=\frac{1}{2}{\left(4\right)}^{x}[/latex]. State the domain, range, and asymptote. https://www.desmos.com/calculator/u8kysdu1wl

Answer: The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].  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 [latex]f\left(x\right)={b}^{x}[/latex] 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 [latex]f\left(x\right)={2}^{x}[/latex], we can then graph the two reflections alongside it. The reflection about the x-axis, [latex]g\left(x\right)={-2}^{x}[/latex], is shown on the left side, and the reflection about the y-axis [latex]h\left(x\right)={2}^{-x}[/latex], is shown on the right side.
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. (a) [latex]g\left(x\right)=-{2}^{x}[/latex] reflects the graph of [latex]f\left(x\right)={2}^{x}[/latex] about the x-axis. (b) [latex]g\left(x\right)={2}^{-x}[/latex] reflects the graph of [latex]f\left(x\right)={2}^{x}[/latex] about the y-axis.

A General Note: Reflections of the Parent Function f(x) = bx

The function [latex]f\left(x\right)=-{b}^{x}[/latex]
  • reflects the parent function [latex]f\left(x\right)={b}^{x}[/latex] about the x-axis.
  • has a y-intercept of [latex]\left(0,-1\right)[/latex].
  • has a range of [latex]\left(-\infty ,0\right)[/latex]
  • has a horizontal asymptote at [latex]y=0[/latex] and domain of [latex]\left(-\infty ,\infty \right)[/latex], which are unchanged from the parent function.
The function [latex]f\left(x\right)={b}^{-x}[/latex]
  • reflects the parent function [latex]f\left(x\right)={b}^{x}[/latex] about the y-axis.
  • has a y-intercept of [latex]\left(0,1\right)[/latex], a horizontal asymptote at [latex]y=0[/latex], a range of [latex]\left(0,\infty \right)[/latex], and a domain of [latex]\left(-\infty ,\infty \right)[/latex], 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, [latex]g\left(x\right)[/latex], that reflects [latex]f\left(x\right)={\left(\frac{1}{4}\right)}^{x}[/latex] about the x-axis. State its domain, range, and asymptote.

Answer: Since we want to reflect the parent function [latex]f\left(x\right)={\left(\frac{1}{4}\right)}^{x}[/latex] about the x-axis, we multiply [latex]f\left(x\right)[/latex] by –1 to get, [latex]g\left(x\right)=-{\left(\frac{1}{4}\right)}^{x}[/latex]. Next we create a table of points.

[latex]x[/latex] –3 –2 –1 0 1 2 3
[latex]g\left(x\right)=-\left(\frac{1}{4}\right)^{x}[/latex] –64 –16 –4 –1 –0.25 –0.0625 –0.0156
Plot the y-intercept, [latex]\left(0,-1\right)[/latex], along with two other points. We can use [latex]\left(-1,-4\right)[/latex] and [latex]\left(1,-0.25\right)[/latex]. 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). The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-\infty ,0\right)[/latex]; the horizontal asymptote is [latex]y=0[/latex].

Try It

Use Desmos to graph the equation for a function, [latex]g\left(x\right)[/latex], that reflects [latex]f\left(x\right)={1.25}^{x}[/latex] about the y-axis. State its domain, range, and asymptote.

Answer: The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(0,\infty \right)[/latex]; the horizontal asymptote is [latex]y=0[/latex]. 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 to arrive at the general equation for translating exponential functions.
Translations of the Parent Function [latex]f\left(x\right)={b}^{x}[/latex]
Translation Form
Shift
  • Horizontally c units to the left
  • Vertically d units up
 [latex]f\left(x\right)={b}^{x+c}+d[/latex]
Stretch and Compress
  • Stretch if |a|>1
  • Compression if 0<|a|<1
 [latex]f\left(x\right)=a{b}^{x}[/latex]
Reflect about the x-axis [latex]f\left(x\right)=-{b}^{x}[/latex]
Reflect about the y-axis [latex]f\left(x\right)={b}^{-x}={\left(\frac{1}{b}\right)}^{x}[/latex]
General equation for all translations [latex]f\left(x\right)=a{b}^{x+c}+d[/latex]

A General Note: Translations of Exponential Functions

A translation of an exponential function has the form [latex-display] f\left(x\right)=a{b}^{x+c}+d[/latex-display] Where the parent function, [latex]y={b}^{x}[/latex], [latex]b>1[/latex], 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.
  • [latex]f\left(x\right)={e}^{x}[/latex] is vertically stretched by a factor of 2, reflected across the y-axis, and then shifted up 4 units.

Answer: We want to find an equation of the general form [latex] f\left(x\right)=a{b}^{x+c}+d[/latex]. We use the description provided to find a, b, c, and d.

  • We are given the parent function [latex]f\left(x\right)={e}^{x}[/latex], 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: [latex]{e}^{-x}[/latex].
  • The graph is shifted vertically 4 units, so d = 4.
Substituting in the general form we get,

[latex]\begin{array} f\left(x\right)\hfill & =a{b}^{x+c}+d\hfill \\ \hfill & =2{e}^{-x+0}+4\hfill \\ \hfill & =2{e}^{-x}+4\hfill \end{array}[/latex]

The domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(4,\infty \right)[/latex]; the horizontal asymptote is [latex]y=4[/latex].

Try It

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.
  • [latex]f\left(x\right)={e}^{x}[/latex] is compressed vertically by a factor of [latex]\frac{1}{3}[/latex], reflected across the x-axis and then shifted down 2 units.

Answer: [latex]f\left(x\right)=-\frac{1}{3}{e}^{x}-2[/latex]; the domain is [latex]\left(-\infty ,\infty \right)[/latex]; the range is [latex]\left(-\infty ,2\right)[/latex]; the horizontal asymptote is [latex]y=2[/latex].

Key Equations

General Form for the Translation of the Parent Function [latex]\text{ }f\left(x\right)={b}^{x}[/latex] [latex]f\left(x\right)=a{b}^{x+c}+d[/latex]

Key Concepts

  • The graph of the function [latex]f\left(x\right)={b}^{x}[/latex] has a y-intercept at [latex]\left(0, 1\right)[/latex], domain [latex]\left(-\infty , \infty \right)[/latex], range [latex]\left(0, \infty \right)[/latex], and horizontal asymptote [latex]y=0[/latex].
  • If [latex]b>1[/latex], the function is increasing. The left tail of the graph will approach the asymptote [latex]y=0[/latex], 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 [latex]y=0[/latex].
  • The equation [latex]f\left(x\right)={b}^{x}+d[/latex] represents a vertical shift of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
  • The equation [latex]f\left(x\right)={b}^{x+c}[/latex] represents a horizontal shift of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
  • Approximate solutions of the equation [latex]f\left(x\right)={b}^{x+c}+d[/latex] can be found using a graphing calculator.
  • The equation [latex]f\left(x\right)=a{b}^{x}[/latex], where [latex]a>0[/latex], represents a vertical stretch if [latex]|a|>1[/latex] or compression if [latex]0<|a|<1[/latex] of the parent function [latex]f\left(x\right)={b}^{x}[/latex].
  • When the parent function [latex]f\left(x\right)={b}^{x}[/latex] is multiplied by –1, the result, [latex]f\left(x\right)=-{b}^{x}[/latex], is a reflection about the x-axis. When the input is multiplied by –1, the result, [latex]f\left(x\right)={b}^{-x}[/latex], is a reflection about the y-axis.
  • All translations of the exponential function can be summarized by the general equation [latex]f\left(x\right)=a{b}^{x+c}+d[/latex].
  • Using the general equation [latex]f\left(x\right)=a{b}^{x+c}+d[/latex], we can write the equation of a function given its description.

Licenses & Attributions

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  • Revision and Adaptation. Provided by: Lumen Learning License: CC BY: Attribution.
  • Characteristics of Graphs of Exponential Functions Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/3pqyjsunkg. License: Public Domain: No Known Copyright.
  • Horizontal and Vertical Translations of Exponential Functions Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/5mrjqegkxk. License: Public Domain: No Known Copyright.
  • Horizontal and Vertical Translations of Exponential Functions 2 Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/rpv1kea0pz. License: Public Domain: No Known Copyright.
  • Horizontal and Vertical Translations of Exponential Functions 3 Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/e5l4eca3ob. License: Public Domain: No Known Copyright.
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  • Solve Exponential Interactive. Authored by: Lumen Learning. Located at: https://www.desmos.com/calculator/lhmpdkbjt0. License: Public Domain: No Known Copyright.

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