4.4 Graphs of logarithmic functions (Page 4/8)

Precalculus

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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 $(- 4, \infty),$ the range $(- \infty, \infty),$ 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.

A General Note

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 $(0, \infty) .$
has range $(- \infty, \infty) .$

How To

Given a logarithmic function with the form $f (x) = \log_{b} (x) + d,$ graph the translation.

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.
Draw the vertical asymptote $x = 0.$
Identify three key points from the parent function. Find new coordinates for the shifted functions by adding $d$ to the $y$ coordinate.
Label the three points.
The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ 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, $(\frac{1}{3}, −1),$ $(1, 0),$ and $(3, 1) .$

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

Label the points $(\frac{1}{3}, −3),$ $(1, −2),$ and $(3, −1) .$

The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ 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 $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

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Try It

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 $(0, \infty),$ the range is $(- \infty, \infty),$ 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, $h (x) = \frac{1}{a} \log_{b} (x) .$ 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.

A General Note

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 $(0, \infty) .$
has range $(- \infty, \infty) .$

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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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4.4 Graphs of logarithmic functions (Page 4/8)

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

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

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

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

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

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

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

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

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

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

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