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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] .

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
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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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