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Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.
(a) g ( x ) = 3 ( 2 ) x stretches the graph of f ( x ) = 2 x vertically by a factor of 3. (b) h ( x ) = 1 3 ( 2 ) x compresses the graph of f ( x ) = 2 x vertically by a factor of 1 3 .

Stretches and compressions of the parent 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 y -intercept of ( 0 , a ) .
  • has a horizontal asymptote at y = 0 , a range of ( 0 , ) , and a domain of ( , ) , which are unchanged from the parent function.

Graphing the stretch of an exponential function

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

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

  • Since b = 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 ) = ( 1 2 ) x will be stretched by a factor of 4.
  • Create a table of points as shown in [link] .
    x 3 2 1 0 1 2 3
    f ( x ) = 4 ( 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 [link] .

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 ( , ) ; the range is ( 0 , ) ; the horizontal asymptote is y = 0.

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Sketch the graph of f ( x ) = 1 2 ( 4 ) x . State the domain, range, and asymptote.

The domain is ( , ) ; the range is ( 0 , ) ; 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).

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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 [link] , and the reflection about the y -axis h ( x ) = 2 x , is shown on the right side of [link] .

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) 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.

Reflections of the parent function f ( x ) = b x

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 ( , 0 )
  • has a horizontal asymptote at y = 0 and domain of ( , ) , 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 , ) , and a domain of ( , ) , which are unchanged from the parent function.

Writing and graphing the reflection of an exponential function

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

Since we want to reflect the parent function f ( x ) = ( 1 4 ) x about the x- axis, we multiply f ( x ) by 1 to get, g ( x ) = ( 1 4 ) x . Next we create a table of points as in [link] .

x 3 2 1 0 1 2 3
g ( x ) = ( 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).

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

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Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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