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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.

The domain is ( , ) ; the range is ( 0 , ) ; 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).
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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 [link] 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 ) = a b x
Reflect about the x -axis f ( x ) = b x
Reflect about the y -axis f ( x ) = b x = ( 1 b ) x
General equation for all translations f ( x ) = a b x + c + d

Translations of exponential functions

A translation of an exponential function has the form

  f ( x ) = a b 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.

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.

We want to find an equation of the general form   f ( x ) = a b x + c + d . We use the description provided to find a , b , c , and 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 ) = a b x + c + d = 2 e x + 0 + 4 = 2 e x + 4

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

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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 1 3 , reflected across the x -axis and then shifted down 2 units.

f ( x ) = 1 3 e x 2 ; the domain is ( , ) ; the range is ( , 2 ) ; the horizontal asymptote is y = 2.

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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 ) = a b x + c + d

Key concepts

  • The graph of the function f ( x ) = b x has a y- intercept at ( 0 ,   1 ) , domain ( ,   ) , range ( 0 ,   ) , and horizontal asymptote y = 0. See [link] .
  • 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 [link] .
  • Approximate solutions of the equation f ( x ) = b x + c + d can be found using a graphing calculator. See [link] .
  • The equation f ( x ) = a b 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 [link] .
  • 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 [link] .
  • All translations of the exponential function can be summarized by the general equation f ( x ) = a b x + c + d . See [link] .
  • Using the general equation f ( x ) = a b x + c + d , we can write the equation of a function given its description. See [link] .

Questions & Answers

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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