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Sketch a graph of f ( x ) = log 3 ( x + 4 ) alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).

The domain is ( 4 , ) , the range ( , ) , and the asymptote x = 4.

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Graphing a vertical shift of y = log b ( x )

When a constant d is added to the parent function f ( x ) = log b ( x ) , the result is a vertical shift     d units in the direction of the sign on d . To visualize vertical shifts, we can observe the general graph of the parent function f ( x ) = log b ( x ) alongside the shift up, g ( x ) = log b ( x ) + d and the shift down, h ( x ) = log b ( x ) d . See [link] .

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=log_b(x)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.

Vertical shifts of the parent function y = log b ( x )

For any constant d , the function f ( x ) = log b ( x ) + d

  • shifts the parent function y = log b ( x ) up d units if d > 0.
  • shifts the parent function y = log b ( x ) down d units if d < 0.
  • has the vertical asymptote x = 0.
  • has domain ( 0 , ) .
  • has range ( , ) .

Given a logarithmic function with the form f ( x ) = log b ( x ) + d , graph the translation.

  1. Identify the vertical shift:
    • If d > 0 , shift the graph of f ( x ) = log b ( x ) up d units.
    • If d < 0 , shift the graph of f ( x ) = log b ( x ) down d units.
  2. Draw the vertical asymptote x = 0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by adding d to the y coordinate.
  4. Label the three points.
  5. The domain is ( 0, ) , the range is ( , ) , and the vertical asymptote is x = 0.

Graphing a vertical shift of the parent function y = log b ( x )

Sketch a graph of f ( x ) = log 3 ( x ) 2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is f ( x ) = log 3 ( x ) 2 , we will notice d = 2. Thus d < 0.

This means we will shift the function f ( x ) = log 3 ( x ) down 2 units.

The vertical asymptote is x = 0.

Consider the three key points from the parent function, ( 1 3 , −1 ) , ( 1 , 0 ) , and ( 3 , 1 ) .

The new coordinates are found by subtracting 2 from the y coordinates.

Label the points ( 1 3 , −3 ) , ( 1 , −2 ) , and ( 3 , −1 ) .

The domain is ( 0 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 1).

The domain is ( 0 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

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Sketch a graph of f ( x ) = log 2 ( x ) + 2 alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Graph of two functions. The parent function is y=log_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).

The domain is ( 0 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

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Graphing stretches and compressions of y = log b ( x )

When the parent function f ( x ) = log b ( x ) is multiplied by a constant a > 0 , the result is a vertical stretch    or compression of the original graph. To visualize stretches and compressions, we set a > 1 and observe the general graph of the parent function f ( x ) = log b ( x ) alongside the vertical stretch, g ( x ) = a log b ( x ) and the vertical compression, h ( x ) = 1 a log b ( x ) . See [link] .

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0  and g(x)=alog_b(x) when a>1 is the translation function with an asymptote at x=0. The graph note the intersection of the two lines at (1, 0). This shows the translation of a vertical stretch.

Vertical stretches and compressions of the parent function y = log b ( x )

For any constant a > 1 , the function f ( x ) = a log b ( x )

  • stretches the parent function y = log b ( x ) vertically by a factor of a if a > 1.
  • compresses the parent function y = log b ( x ) vertically by a factor of a if 0 < a < 1.
  • has the vertical asymptote x = 0.
  • has the x -intercept ( 1 , 0 ) .
  • has domain ( 0 , ) .
  • has range ( , ) .

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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