4.4 Graphs of logarithmic functions (Page 4/8)

Precalculus

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Sketch a graph of $f (x) = \log_{3} (x + 4)$ alongside its parent function. Include the key points and asymptotes on the graph. State the domain, range, and asymptote.

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1, 0), and (3, 1).The translation function f(x)=log_3(x+4) has an asymptote at x=-4 and labeled points at (-3, 0) and (-1, 1).

The domain is $(- 4, \infty),$ the range $(- \infty, \infty),$ and the asymptote $x = - 4.$

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Graphing a vertical shift of y = log _b ( x )

When a constant $d$ is added to the parent function $f (x) = \log_{b} (x),$ the result is a vertical shift $d$ units in the direction of the sign on $d .$ To visualize vertical shifts, we can observe the general graph of the parent function $f (x) = \log_{b} (x)$ alongside the shift up, $g (x) = \log_{b} (x) + d$ and the shift down, $h (x) = \log_{b} (x) - d .$ See [link] .

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)+d is the translation function with an asymptote at x=0. This shows the translation of shifting up. 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)-d is the translation function with an asymptote at x=0. This shows the translation of shifting down.

A General Note

Vertical shifts of the parent function y = log _b ( x )

For any constant $d,$ the function $f (x) = \log_{b} (x) + d$

shifts the parent function $y = \log_{b} (x)$ up $d$ units if $d > 0.$
shifts the parent function $y = \log_{b} (x)$ down $d$ units if $d < 0.$
has the vertical asymptote $x = 0.$
has domain $(0, \infty) .$
has range $(- \infty, \infty) .$

How To

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

Identify the vertical shift:
- If $d > 0,$ shift the graph of $f (x) = \log_{b} (x)$ up $d$ units.
- If $d < 0,$ shift the graph of $f (x) = \log_{b} (x)$ down $d$ units.
Draw the vertical asymptote $x = 0.$
Identify three key points from the parent function. Find new coordinates for the shifted functions by adding $d$ to the $y$ coordinate.
Label the three points.
The domain is $(0, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 0.$

Graphing a vertical shift of the parent function y = log _b ( x )

Sketch a graph of $f (x) = \log_{3} (x) - 2$ alongside its parent function. Include the key points and asymptote on the graph. State the domain, range, and asymptote.

Since the function is $f (x) = \log_{3} (x) - 2,$ we will notice $d = - 2.$ Thus $d < 0.$

This means we will shift the function $f (x) = \log_{3} (x)$ down 2 units.

The vertical asymptote is $x = 0.$

Consider the three key points from the parent function, $(\frac{1}{3}, −1),$ $(1, 0),$ and $(3, 1) .$

The new coordinates are found by subtracting 2 from the y coordinates.

Label the points $(\frac{1}{3}, −3),$ $(1, −2),$ and $(3, −1) .$

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

Graph of two functions. The parent function is y=log_3(x), with an asymptote at x=0 and labeled points at (1/3, -1), (1, 0), and (3, 1).The translation function f(x)=log_3(x)-2 has an asymptote at x=0 and labeled points at (1, 0) and (3, 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 $f (x) = \log_{2} (x) + 2$ 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_2(x), with an asymptote at x=0 and labeled points at (1, 0), and (2, 1).The translation function f(x)=log_2(x)+2 has an asymptote at x=0 and labeled points at (0.25, 0) and (0.5, 1).

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

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Graphing stretches and compressions of y = log _b ( x )

When the parent function $f (x) = \log_{b} (x)$ is multiplied by a constant $a > 0,$ the result is a vertical stretch or compression of the original graph. To visualize stretches and compressions, we set $a > 1$ and observe the general graph of the parent function $f (x) = \log_{b} (x)$ alongside the vertical stretch, $g (x) = a \log_{b} (x)$ and the vertical compression, $h (x) = \frac{1}{a} \log_{b} (x) .$ See [link] .

Graph of two functions. The parent function is f(x)=log_b(x), with an asymptote at x=0 and g(x)=alog_b(x) when a>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 vertical stretch.

A General Note

Vertical stretches and compressions of the parent function y = log _b ( x )

For any constant $a > 1,$ the function $f (x) = a \log_{b} (x)$

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 $0 < a < 1.$
has the vertical asymptote $x = 0.$
has the x -intercept $(1, 0) .$
has domain $(0, \infty) .$
has range $(- \infty, \infty) .$

Questions & Answers

how did you get 1640

Noor Reply

If auger is pair are the roots of equation x2+5x-3=0

Peter Reply

Wayne and Dennis like to ride the bike path from Riverside Park to the beach. Dennis’s speed is seven miles per hour faster than Wayne’s speed, so it takes Wayne 2 hours to ride to the beach while it takes Dennis 1.5 hours for the ride. Find the speed of both bikers.

MATTHEW Reply

420

Sharon

from theory: distance [miles] = speed [mph] × time [hours] info #1 speed_Dennis × 1.5 = speed_Wayne × 2 => speed_Wayne = 0.75 × speed_Dennis (i) info #2 speed_Dennis = speed_Wayne + 7 [mph] (ii) use (i) in (ii) => [...] speed_Dennis = 28 mph speed_Wayne = 21 mph

George

Let W be Wayne's speed in miles per hour and D be Dennis's speed in miles per hour. We know that W + 7 = D and W * 2 = D * 1.5. Substituting the first equation into the second: W * 2 = (W + 7) * 1.5 W * 2 = W * 1.5 + 7 * 1.5 0.5 * W = 7 * 1.5 W = 7 * 3 or 21 W is 21 D = W + 7 D = 21 + 7 D = 28

Salma

Devon is 32 32 years older than his son, Milan. The sum of both their ages is 54 54. Using the variables d d and m m to represent the ages of Devon and Milan, respectively, write a system of equations to describe this situation. Enter the equations below, separated by a comma.

Aaron Reply

find product (-6m+6) ( 3m²+4m-3)

SIMRAN Reply

-42m²+60m-18

Salma

what is the solution

bill

how did you arrive at this answer?

bill

-24m+3+3mÁ^2

Susan

i really want to learn

Amira

I only got 42 the rest i don't know how to solve it. Please i need help from anyone to help me improve my solving mathematics please

Amira

Hw did u arrive to this answer.

Aphelele

Bajemah

-6m(3mA²+4m-3)+6(3mA²+4m-3) =-18m²A²-24m²+18m+18mA²+24m-18 Rearrange like items -18m²A²-24m²+42m+18A²-18

Salma

complete the table of valuesfor each given equatio then graph. 1.x+2y=3

Jovelyn Reply

x=3-2y

Salma

y=x+3/2

Salma

Enock

given that (7x-5):(2+4x)=8:7find the value of x

Nandala

3x-12y=18

Kelvin

please why isn't that the 0is in ten thousand place

Grace Reply

please why is it that the 0is in the place of ten thousand

Grace

Send the example to me here and let me see

Stephen

A meditation garden is in the shape of a right triangle, with one leg 7 feet. The length of the hypotenuse is one more than the length of one of the other legs. Find the lengths of the hypotenuse and the other leg

Marry Reply

how far

Abubakar

cool u

Enock

state in which quadrant or on which axis each of the following angles given measure. in standard position would lie 89°

Abegail Reply

hello

BenJay

Method

I am eliacin, I need your help in maths

Rood

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Sir

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Amoon

however, may I ask you some questions about Algarba?

Amoon

Enock

what the last part of the problem mean?

Roger

The Jones family took a 15 mile canoe ride down the Indian River in three hours. After lunch, the return trip back up the river took five hours. Find the rate, in mph, of the canoe in still water and the rate of the current.

cameron Reply

Shakir works at a computer store. His weekly pay will be either a fixed amount, $925, or $500 plus 12% of his total sales. How much should his total sales be for his variable pay option to exceed the fixed amount of $925.

mahnoor Reply

I'm guessing, but it's somewhere around $4335.00 I think

Lewis

12% of sales will need to exceed 925 - 500, or 425 to exceed fixed amount option. What amount of sales does that equal? 425 ÷ (12÷100) = 3541.67. So the answer is sales greater than 3541.67. Check: Sales = 3542 Commission 12%=425.04 Pay = 500 + 425.04 = 925.04. 925.04 > 925.00

Munster

difference between rational and irrational numbers

Arundhati Reply

When traveling to Great Britain, Bethany exchanged $602 US dollars into £515 British pounds. How many pounds did she receive for each US dollar?

Jakoiya Reply

how to reduced echelon form

Solomon Reply

Jazmine trained for 3 hours on Saturday. She ran 8 miles and then biked 24 miles. Her biking speed is 4 mph faster than her running speed. What is her running speed?

Zack Reply

d=r×t the equation would be 8/r+24/r+4=3 worked out

Sheirtina

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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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4.4 Graphs of logarithmic functions (Page 4/8)

Graphing a vertical shift of y = log b ( x )

Vertical shifts of the parent function y = log b ( x )

Graphing a vertical shift of the parent function y = log b ( x )

Graphing stretches and compressions of y = log b ( x )

Vertical stretches and compressions of the parent function y = log b ( x )

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Graphing a vertical shift of y = log _b ( x )

Vertical shifts of the parent function y = log _b ( x )

Graphing a vertical shift of the parent function y = log _b ( x )

Graphing stretches and compressions of y = log _b ( x )

Vertical stretches and compressions of the parent function y = log _b ( x )