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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.

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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.

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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 , ) . 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.

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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 , ) , the range is ( , ) , 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.

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

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
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply

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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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