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Translations of the Parent Function y = log b ( x )
Translation Form
Shift
  • Horizontally c units to the left
  • Vertically d units up
y = log b ( x + c ) + d
Stretch and Compress
  • Stretch if | a | > 1
  • Compression if | a | < 1
y = a log b ( x )
Reflect about the x -axis y = log b ( x )
Reflect about the y -axis y = log b ( x )
General equation for all translations y = a log b ( x + c ) + d

Translations of logarithmic functions

All translations of the parent logarithmic function, y = log b ( x ) , have the form

  f ( x ) = a log b ( x + c ) + d

where the parent function, y = log b ( x ) , b > 1 , is

  • shifted vertically up d units.
  • shifted horizontally to the left c units.
  • stretched vertically by a factor of | a | if | a | > 0.
  • compressed vertically by a factor of | a | if 0 < | a | < 1.
  • reflected about the x- axis when a < 0.

For f ( x ) = log ( x ) , the graph of the parent function is reflected about the y -axis.

Finding the vertical asymptote of a logarithm graph

What is the vertical asymptote of f ( x ) = −2 log 3 ( x + 4 ) + 5 ?

The vertical asymptote is at x = 4.

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What is the vertical asymptote of f ( x ) = 3 + ln ( x 1 ) ?

x = 1

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Finding the equation from a graph

Find a possible equation for the common logarithmic function graphed in [link] .

Graph of a logarithmic function with a vertical asymptote at x=-2, has been vertically reflected, and passes through the points (-1, 1) and (2, -1).

This graph has a vertical asymptote at x = –2 and has been vertically reflected. We do not know yet the vertical shift or the vertical stretch. We know so far that the equation will have form:

f ( x ) = a log ( x + 2 ) + k

It appears the graph passes through the points ( –1 , 1 ) and ( 2 , –1 ) . Substituting ( –1 , 1 ) ,

1 = a log ( −1 + 2 ) + k Substitute  ( −1 , 1 ) . 1 = a log ( 1 ) + k Arithmetic . 1 = k log(1) = 0.

Next, substituting in ( 2 , –1 ) ,

1 = a log ( 2 + 2 ) + 1 Plug in  ( 2 , −1 ) . 2 = a log ( 4 ) Arithmetic .    a = 2 log ( 4 ) Solve for  a .

This gives us the equation f ( x ) = 2 log ( 4 ) log ( x + 2 ) + 1.

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Give the equation of the natural logarithm graphed in [link] .

Graph of a logarithmic function with a vertical asymptote at x=-3, has been vertically stretched by 2, and passes through the points (-1, -1).

f ( x ) = 2 ln ( x + 3 ) 1

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Is it possible to tell the domain and range and describe the end behavior of a function just by looking at the graph?

Yes, if we know the function is a general logarithmic function. For example, look at the graph in [link] . The graph approaches x = −3 (or thereabouts) more and more closely, so x = −3 is, or is very close to, the vertical asymptote. It approaches from the right, so the domain is all points to the right, { x | x > −3 } . The range, as with all general logarithmic functions, is all real numbers. And we can see the end behavior because the graph goes down as it goes left and up as it goes right. The end behavior is that as x 3 + , f ( x ) and as x , f ( x ) .

Access these online resources for additional instruction and practice with graphing logarithms.

Key equations

General Form for the Translation of the Parent Logarithmic Function   f ( x ) = log b ( x )   f ( x ) = a log b ( x + c ) + d

Key concepts

  • To find the domain of a logarithmic function, set up an inequality showing the argument greater than zero, and solve for x . See [link] and [link]
  • The graph of the parent function f ( x ) = log b ( x ) has an x- intercept at ( 1 , 0 ) , domain ( 0 , ) , range ( , ) , vertical asymptote x = 0 , and
    • if b > 1 , the function is increasing.
    • if 0 < b < 1 , the function is decreasing.
    See [link] .
  • The equation f ( x ) = log b ( x + c ) shifts the parent function y = log b ( x ) horizontally
    • left c units if c > 0.
    • right c units if c < 0.
    See [link] .
  • The equation f ( x ) = log b ( x ) + d shifts the parent function y = log b ( x ) vertically
    • up d units if d > 0.
    • down d units if d < 0.
    See [link] .
  • For any constant a > 0 , the equation 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 | a | < 1.
    See [link] and [link] .
  • When the parent function y = 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.
    • The equation f ( x ) = log b ( x ) represents a reflection of the parent function about the x- axis.
    • The equation f ( x ) = log b ( x ) represents a reflection of the parent function about the y- axis.
    See [link] .
    • A graphing calculator may be used to approximate solutions to some logarithmic equations See [link] .
  • All translations of the logarithmic function can be summarized by the general equation   f ( x ) = a log b ( x + c ) + d . See [link] .
  • Given an equation with the general form   f ( x ) = a log b ( x + c ) + d , we can identify the vertical asymptote x = c for the transformation. See [link] .
  • Using the general equation f ( x ) = a log b ( x + c ) + d , we can write the equation of a logarithmic function given its graph. See [link] .

Questions & Answers

The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris

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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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