6.4 Graphs of logarithmic functions (Page 5/8)

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How To

Given a logarithmic function with the form $f (x) = a \log_{b} (x),$ $a > 0,$ graph the translation.

Identify the vertical stretch or compressions:
- If $| a | > 1,$ the graph of $f (x) = \log_{b} (x)$ is stretched by a factor of $a$ units.
- If $| a | < 1,$ the graph of $f (x) = \log_{b} (x)$ is compressed by a factor of $a$ units.
Draw the vertical asymptote $x = 0.$
Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the $y$ coordinates by $a .$
Label the three points.
The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Graphing a stretch or compression of the parent function y = log _b ( x )

Sketch a graph of $f (x) = 2 \log_{4} (x)$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is $f (x) = 2 \log_{4} (x),$ we will notice $a = 2.$

This means we will stretch the function $f (x) = \log_{4} (x)$ by a factor of 2.

The vertical asymptote is $x = 0.$

Consider the three key points from the parent function, $(\frac{1}{4}, −1),$ $(1, 0),$ and $(4, 1) .$

The new coordinates are found by multiplying the $y$ coordinates by 2.

Label the points $(\frac{1}{4}, −2),$ $(1, 0),$ and $(4, 2) .$

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

Graph of two functions. The parent function is y=log_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=2log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (2, 1).

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

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Sketch a graph of $f (x) = \frac{1}{2} \log_{4} (x)$ 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_4(x), with an asymptote at x=0 and labeled points at (1, 0), and (4, 1).The translation function f(x)=(1/2)log_4(x) has an asymptote at x=0 and labeled points at (1, 0) and (16, 1).

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

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Combining a shift and a stretch

Sketch a graph of $f (x) = 5 \log (x + 2) .$ State the domain, range, and asymptote.

Remember: what happens inside parentheses happens first. First, we move the graph left 2 units, then stretch the function vertically by a factor of 5, as in [link] . The vertical asymptote will be shifted to $x = −2.$ The x -intercept will be $(−1, 0) .$ The domain will be $(−2, \infty) .$ Two points will help give the shape of the graph: $(−1, 0)$ and $(8, 5).$ We chose $x = 8$ as the x -coordinate of one point to graph because when $x = 8,$ $x + 2 = 10,$ the base of the common logarithm.

Graph of three functions. The parent function is y=log(x), with an asymptote at x=0. The first translation function y=5log(x+2) has an asymptote at x=-2. The second translation function y=log(x+2) has an asymptote at x=-2.

The domain is $(- 2, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = - 2.$

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Sketch a graph of the function $f (x) = 3 \log (x - 2) + 1.$ State the domain, range, and asymptote.

Graph of f(x)=3log(x-2)+1 with an asymptote at x=2.

The domain is $(2, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 2.$

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Graphing reflections of f ( x ) = log _b ( x )

When the parent function $f (x) = \log_{b} (x)$ is multiplied by $−1,$ the result is a reflection about the x -axis. When the input is multiplied by $−1,$ the result is a reflection about the y -axis. To visualize reflections, we restrict $b > 1,$ and observe the general graph of the parent function $f (x) = \log_{b} (x)$ alongside the reflection about the x -axis, $g (x) = {−log}_{b} (x)$ and the reflection about the y -axis, $h (x) = \log_{b} (- x) .$

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) when b>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 reflection about the x-axis.

A General Note

Reflections of the parent function y = log _b ( x )

The function $f (x) = {−log}_{b} (x)$

reflects the parent function $y = \log_{b} (x)$ about the x -axis.
has domain, $(0, \infty),$ range, $(- \infty, \infty),$ and vertical asymptote, $x = 0,$ which are unchanged from the parent function.

The function $f (x) = \log_{b} (- x)$

reflects the parent function $y = \log_{b} (x)$ about the y -axis.
has domain $(- \infty, 0) .$
has range, $(- \infty, \infty),$ and vertical asymptote, $x = 0,$ which are unchanged from the parent function.

Questions & Answers

If c is the cost function for a particular product, find the marginal cost functions and their values at x=10 a. c(x) = 800+ 0.04x + 0.0002x² b. c(x) = 250 + 100x + 0.001x²

Mamush Reply

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is there an error on the one about the dime's thickness? says 2.2x10⁶=0.00135 m

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how to reduce an equation?

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by manipulation of both side

9(y+8)-27 is 9y+45. Why can't you reduce that to y+5? I know that's wrong but can't explain why

Patrick Reply

when you reduce an equation to its simplest terms, you can't change the value of the equation. reducing it to y + 5 is equivalent to dividing it by 9 which changes the value. you can multiply it by 1 or 9/9 which would give 9(y + 5). multiplying it by one does not change the value.

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Definition of economics according to karl Marx Thomas malthus Jeremy bentham David Ricardo J.K

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_3_2_1

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The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu

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1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3

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Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22

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Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

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Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113

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Source: OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3

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6.4 Graphs of logarithmic functions (Page 5/8)

Graphing a stretch or compression of the parent function y = log b ( x )

Combining a shift and a stretch

Graphing reflections of f ( x ) = log b ( x )

Reflections of the parent function y = log b ( x )

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Graphing a stretch or compression of the parent function y = log _b ( x )

Graphing reflections of f ( x ) = log _b ( x )

Reflections of the parent function y = log _b ( x )