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4.2 Graphs of exponential functions  (Page 5/6)

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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] .
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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