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

summarize halerambos & holbon

David Reply

the Three stages of Auguste Comte

Clementina Reply

what are agents of socialization

Antonio Reply

sociology of education

Nuhu Reply

definition of sociology of education

Nuhu

what is culture

Abdulrahim Reply

shared beliefs, values, and practices

AI-Robot

What are the two type of scientific method

ogunniran Reply

I'm willing to join you

Aceng Reply

what are the scientific method of sociology

Man

what is socialization

ogunniran Reply

the process wherein people come to understand societal norms and expectations, to accept society's beliefs, and to be aware of societal values

AI-Robot

scientific method in doing research

ogunniran

defimition of sickness in afica

Anita

Cosmology

ogunniran

Hmmm

ogunniran

list and explain the terms that found in society

REMMY Reply

list and explain the terms that found in society

Mukhtar

what are the agents of socialization

Antonio

Family Peer group Institution

Abdulwajud

I mean the definition

Antonio

ways of perceived deviance indifferent society

Naomi Reply

reasons of joining groups

SAM

to bring development to the nation at large

Hyellafiya

entails of consultative and consensus building from others

Gadama

World first Sociologist?

Abu

What is evolutionary model

Muhammad Reply

Evolution models refer to mathematical and computational representations of the processes involved in biological evolution. These models aim to simulate and understand how species change over time through mechanisms such as natural selection, genetic drift, and mutation. Evolutionary models can be u

faruk

what are the modern trends in religious behaviours

Selekeye Reply

what are social norms

Daniel Reply

shared standards of acceptable behavior by the group or appropriate behavior in a particular institution or those behaviors that are acceptable in a society

Lucius

that is how i understood it

Lucius

examples of societal norms

Diamond

Discuss the characteristics of the research located within positivist and the interpretivist paradigm

Tariro Reply

what is Industrialisation

Selekeye Reply

industrialization

Angelo

Got questions? Join the online conversation and get instant answers!

Jobilize.com Reply

<< 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
	1 Lec:1 Descriptive Epidemiology By Janet Forrester Start Quiz
	Cultural Anthropology Assignment 2 By Richley Crapo Start Assignment
	18 AP Key Terms 18 Cardiovascular System Blood By OpenStax Start Key Terms
	Tournament Director Quiz By Eddie Unverzagt Start Quiz
	7 Neuroanatomy 07 The Visual System By Stephen Voron Start Quiz
	42 Biology 42 The Immune System MCQ By OpenStax Start Quiz
	2011 Dynamics CRM By Danielrosenberger Start Quiz
©flickr: eLife	Mycobacterial Infections (Mandell 8th Ch 251-254) By Cath Yu Start Quiz
	Business Law Ethics By Kevin Moquin Start Quiz
©flickr: Elliott	Spanish Verbs Subject Pronouns By Mariah Hauptman 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 )