<< Chapter < Page Chapter >> Page >

Given a logarithmic function with the form f ( x ) = a log b ( x ) , a > 0 , graph the translation.

  1. 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.
  2. Draw the vertical asymptote x = 0.
  3. Identify three key points from the parent function. Find new coordinates for the shifted functions by multiplying the y coordinates by a .
  4. Label the three points.
  5. The domain is ( 0 , ) , the range is ( , ) , 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, ( 1 4 , −1 ) , ( 1 , 0 ), and ( 4 , 1 ) .

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

Label the points ( 1 4 , −2 ) , ( 1 , 0 ) , and ( 4 , 2 ) .

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Sketch a graph of f ( x ) = 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 , ) , the range is ( , ) , and the vertical asymptote is x = 0.

Got questions? Get instant answers now!

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 , ) . 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 , ) , the range is ( , ) , and the vertical asymptote is x = 2.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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 , ) , the range is ( , ) , and the vertical asymptote is x = 2.

Got questions? Get instant answers now!

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.

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 , ) , range, ( , ) , 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 ( , 0 ) .
  • has range, ( , ) , and vertical asymptote, x = 0 , which are unchanged from the parent function.

Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask