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4.4 Graphs of logarithmic functions  (Page 4/8)

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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   ( − ∞ , ∞ ) .

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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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