<< 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

bsc F. y algebra and trigonometry pepper 2
Aditi Reply
given that x= 3/5 find sin 3x
Adamu Reply
4
DB
remove any signs and collect terms of -2(8a-3b-c)
Joeval Reply
-16a+6b+2c
Will
is that a real answer
Joeval
(x2-2x+8)-4(x2-3x+5)
Ayush Reply
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
X2-2X+8-4X2+12X-20=0 (X2-4X2)+(-2X+12X)+(-20+8)= 0 -3X2+10X-12=0 3X2-10X+12=0 Use quadratic formula To find the answer answer (5±Root11i)/3
master
Soo sorry (5±Root11* i)/3
master
x2-2x+8-4x2+12x-20 x2-4x2-2x+12x+8-20 -3x2+10x-12 now you can find the answer using quadratic
Mukhtar
explain and give four example of hyperbolic function
Lukman Reply
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
Racelle Reply
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Anurag Reply
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
Jhovie Reply
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
Charmaine Reply
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
Snalo Reply
what are you up to?
Mark Reply
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
Propessor Reply
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
sita Reply
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
Pearl Reply
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
haha. already finished college
Jeffrey
how about you? what grade are you now?
Jeffrey
I'm going to 11grade
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
Nelson Reply
which equation
Miranda
I dont know. lol
Jeffrey
please where is the equation
Miranda
ask nelson. lol
Jeffrey

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask