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4.2 Graphs of exponential functions  (Page 3/6)

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

For any constants   c   and   d , the function   f ( x ) = b x + c + d   shifts the parent function   f ( x ) = b x

  • 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   ( 0 , b c + d ) .
  • The horizontal asymptote becomes   y = d .
  • The range becomes   ( d , ∞ ) .
  • The domain,   ( − ∞ , ∞ ) , remains unchanged.

Given an exponential function with the form   f ( x ) = b x + c + d , graph the translation.

  1. Draw the horizontal asymptote   y = d .
  2. Identify the shift as   ( − c , d ) .   Shift the graph of   f ( x ) = b x   left   c   units if   c   is positive, and right   c   units if c   is negative.
  3. Shift the graph of   f ( x ) = b x   up   d   units if   d   is positive, and down   d   units if   d   is negative.
  4. State the domain,   ( − ∞ , ∞ ) , the range,   ( d , ∞ ) , and the horizontal asymptote   y = d .

Graphing a shift of an exponential function

Graph   f ( x ) = 2 x + 1 − 3.   State the domain, range, and asymptote.

We have an exponential equation of the form   f ( x ) = b x + c + d , with   b = 2 ,   c = 1 , and   d = − 3.

Draw the horizontal asymptote   y = d , so draw   y = −3.

Identify the shift as   ( − c , d ) , so the shift is   ( − 1 , −3 ) .

Shift the graph of   f ( x ) = b x   left 1 units and down 3 units.

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   ( − ∞ , ∞ ) ;   the range is   ( − 3 , ∞ ) ;   the horizontal asymptote is   y = −3.

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Graph   f ( x ) = 2 x − 1 + 3.   State domain, range, and asymptote.

The domain is   ( − ∞ , ∞ ) ;   the range is   ( 3 , ∞ ) ;   the horizontal asymptote is   y = 3.

Graph of the function, f(x) = 2^(x-1)+3, with an asymptote at y=3. Labeled points in the graph are (-1, 3.25), (0, 3.5), and (1, 4).
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Given an equation of the form   f ( x ) = b x + c + d   for   x , use a graphing calculator to approximate the solution.

  • Press [Y=] . Enter the given exponential equation in the line headed “ Y 1 = ”.
  • Enter the given value for   f ( x )   in the line headed “ Y 2 = ”.
  • Press [WINDOW] . Adjust the y -axis so that it includes the value entered for “ Y 2 = ”.
  • Press [GRAPH] to observe the graph of the exponential function along with the line for the specified value of   f ( x ) .
  • To find the value of   x , we compute the point of intersection. Press [2ND] then [CALC] . Select “intersect” and press [ENTER] three times. The point of intersection gives the value of x for the indicated value of the function.

Approximating the solution of an exponential equation

Solve   42 = 1.2 ( 5 ) x + 2.8   graphically. Round to the nearest thousandth.

Press [Y=] and enter   1.2 ( 5 ) x + 2.8   next to Y 1 =. Then enter 42 next to Y2= . For a window, use the values –3 to 3 for   x   and –5 to 55 for   y .   Press [GRAPH] . The graphs should intersect somewhere near   x = 2.

For a better approximation, press [2ND] then [CALC] . Select [5: intersect] and press [ENTER] three times. The x -coordinate of the point of intersection is displayed as 2.1661943. (Your answer may be different if you use a different window or use a different value for Guess? ) To the nearest thousandth,   x ≈ 2.166.

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Solve   4 = 7.85 ( 1.15 ) x − 2.27   graphically. Round to the nearest thousandth.

x ≈ − 1.608

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Graphing a stretch or compression

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   f ( x ) = b x   by a constant   | a | > 0.   For example, if we begin by graphing the parent function   f ( x ) = 2 x , we can then graph the stretch, using   a = 3 , to get   g ( x ) = 3 ( 2 ) x   as shown on the left in [link] , and the compression, using   a = 1 3 , to get   h ( x ) = 1 3 ( 2 ) x   as shown on the right in [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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