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학습 가이드 > College Algebra

Stretch, Compress, or Reflect an Exponential Function

Learning Objectives

  • Graph a stretched or compressed exponential function
  • Graph a reflected exponential function
  • Write the equation of an exponential function that has been transformed
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].

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  • Question ID 129498. Authored by: Day, Alyson. License: CC BY: Attribution. License terms: IMathAS Community LicenseCC-BY + GPL.
  • College Algebra. Provided by: OpenStax Authored by: Abramson, Jay et al.. Located at: https://openstax.org/books/college-algebra/pages/1-introduction-to-prerequisites. License: CC BY: Attribution. License terms: Download for free at http://cnx.org/contents/[email protected].