6.4 Graphs of logarithmic functions (Page 4/8)

Page 4 / 8

Try It

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 $x = - 4.$

Got questions? Get instant answers now!

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 $x = 0.$

Got questions? Get instant answers now!

Try It

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 $x = 0.$

Got questions? Get instant answers now!

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

Why is b in the answer

Dahsolar Reply

how do you work it out?

Brad Reply

answer

Ernest

heheheehe

Nitin

(Pcos∅+qsin∅)/(pcos∅-psin∅)

John Reply

how to do that?

Rosemary Reply

what is it about?

Amoah

how to answer the activity

Chabelita Reply

how to solve the activity

Chabelita

solve for X,,4^X-6(2^)-16=0

Alieu Reply

x4xminus 2

Lominate

sobhan Singh jina uniwarcity tignomatry ka long answers tile questions

harish Reply

t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080

ZAHRO Reply

If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .

Swetha Reply

Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?

Patrick Reply

Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?

Patrick

how I can simplify algebraic expressions

Katleho Reply

Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?

Mary Reply

23yrs

Yeboah

lairenea's age is 23yrs

ACKA

Katleho

Ello everyone

Katleho

Laurene is 46 yrs and Mae is 23 is

Solomon

hey people

christopher

age does not matter

christopher

solve for X, 4^x-6(2*)-16=0

Alieu

prove`x^3-3x-2cosA=0 (-π<A<=π

Mayank Reply

create a lesson plan about this lesson

Rose Reply

Excusme but what are you wrot?

<< Chapter < Page Page > Chapter >>

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

100% Free Mobile Applications
Receive real-time job alerts and never miss the right job again

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

	6 Graphing stretches and compressions of y = log b ( x ) By OpenStax Read Online Course
	4 Graphing a horizontal shift of f ( x ) = log b ( x ) By OpenStax Read Online Course
	10 Psychology MCQ 2011 Final Exam By John Gabrieli Start Exam
	Clinical Issues in TB Management By Cath Yu Start Quiz
	2 BOD - Neuropathology By Brooke Delaney Start Exam
	Interviewing Skills MCQ By Mary Matera Start Quiz
	5 Neuroanatomy 05 Somesthetic Sensation By Stephen Voron Start Quiz
©flickr: Clive	Sports Psych. Research Methods By Dakota Bocan Start Flashcards
	21 Sociology 21 Social Movements Social Change MCQ By OpenStax Start Quiz
	3 AP 03 Cellular Level of Organization MCQ By OpenStax Start Quiz
	6 Physiotherapy Flashcards Set 6 By Rhodes Start Flashcards
	8 Arts Society: Theater 8 By Jonathan Long Start Quiz

6.4 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 )