6.4 Graphs of logarithmic functions (Page 3/8)

<< Chapter < Page

Algebra and trigonometry

Page 3 / 8

Chapter >> Page >

How To

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

Draw and label the vertical asymptote, $x = 0.$
Plot the x- intercept, $(1, 0) .$
Plot the key point $(b, 1) .$
Draw a smooth curve through the points.
State the domain, $(0, \infty),$ the range, $(- \infty, \infty),$ and the vertical asymptote, $x = 0.$

Graphing a logarithmic function with the form f ( x ) = log _b ( x ).

Graph $f (x) = \log_{5} (x) .$ State the domain, range, and asymptote.

Before graphing, identify the behavior and key points for the graph.

Since $b = 5$ is greater than one, we know the function is increasing. The left tail of the graph will approach the vertical asymptote $x = 0,$ and the right tail will increase slowly without bound.
The x -intercept is $(1, 0) .$
The key point $(5, 1)$ is on the graph.
We draw and label the asymptote, plot and label the points, and draw a smooth curve through the points (see [link] ).

Graph of f(x)=log_5(x) with labeled points at (1, 0) and (5, 1). The y-axis is the asymptote.

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

Graph $f (x) = \log_{\frac{1}{5}} (x) .$ State the domain, range, and asymptote.

Graph of f(x)=log_(1/5)(x) with labeled points at (1/5, 1) and (1, 0). The y-axis is the asymptote.

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

Got questions? Get instant answers now!

Graphing transformations of logarithmic functions

As we mentioned in the beginning of the section, transformations of logarithmic graphs behave similarly to those of other parent functions. We can shift, stretch, compress, and reflect the parent function $y = \log_{b} (x)$ without loss of shape.

Graphing a horizontal shift of f ( x ) = log _b ( x )

When a constant $c$ is added to the input of the parent function $f (x) = l o g_{b} (x),$ the result is a horizontal shift $c$ units in the opposite direction of the sign on $c .$ To visualize horizontal shifts, we can observe the general graph of the parent function $f (x) = \log_{b} (x)$ and for $c > 0$ alongside the shift left, $g (x) = \log_{b} (x + c),$ and the shift right, $h (x) = \log_{b} (x - c) .$ 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+c) is the translation function with an asymptote at x=-c. This shows the translation of shifting left.

A General Note

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

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

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

How To

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

Identify the horizontal shift:
1. If $c > 0,$ shift the graph of $f (x) = \log_{b} (x)$ left $c$ units.
2. If $c < 0,$ shift the graph of $f (x) = \log_{b} (x)$ right $c$ units.
Draw the vertical asymptote $x = - c .$
Identify three key points from the parent function. Find new coordinates for the shifted functions by subtracting $c$ from the $x$ coordinate.
Label the three points.
The Domain is $(- c, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = - c .$

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

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

Since the function is $f (x) = \log_{3} (x - 2),$ we notice $x + (- 2) = x - 2.$

Thus $c = - 2,$ so $c < 0.$ This means we will shift the function $f (x) = \log_{3} (x)$ right 2 units.

The vertical asymptote is $x = - (- 2)$ or $x = 2.$

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

The new coordinates are found by adding 2 to the $x$ coordinates.

Label the points $(\frac{7}{3}, −1),$ $(3, 0),$ and $(5, 1) .$

The domain is $(2, \infty),$ the range is $(- \infty, \infty),$ and the vertical asymptote is $x = 2.$

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=2 and labeled points at (3, 0) and (5, 1).

Got questions? Get instant answers now!

Questions & Answers

What are types of cell

Nansoh Reply

how can I get this book

Gatyin Reply

what is lump

Chineye Reply

what is cell

Maluak Reply

what is biology

Maluak

what's cornea?

Majak Reply

what are cell

Achol

Explain the following terms . (1) Abiotic factors in an ecosystem

Nomai Reply

Abiotic factors are non living components of ecosystem.These include physical and chemical elements like temperature,light,water,soil,air quality and oxygen etc

Qasim

what is biology

daniel Reply

what is diffusion

Emmanuel Reply

passive process of transport of low-molecular weight material according to its concentration gradient

AI-Robot

what is production?

Catherine

Pathogens and diseases

how did the oxygen help a human being

Achol Reply

how did the nutrition help the plants

Achol Reply

Biology is a branch of Natural science which deals/About living Organism.

Ahmedin Reply

what is phylogeny

Odigie Reply

evolutionary history and relationship of an organism or group of organisms

AI-Robot

Deng

what is biology

Hajah Reply

cell is the smallest unit of the humanity biologically

Abraham

Achol

what is biology

Victoria Reply

what is biology

Abraham

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

	4 Graphing a horizontal shift of f ( x ) = log b ( x ) By OpenStax Read Online Course
	2 Graphing logarithmic functions By OpenStax Read Online Course
	14 Dr Landholt Large Animal Medicine-GI quiz By Brooke Delaney Start Exam
	Chemistry Practice Test By Sandhills MLT Start Test
	Financial Markets By Keyaira Braxton Start Exam
©flickr: Aaron	Computer Literacy Exam 1 By Lakeima Roberts Start Quiz
	11 Sociology 11 Race and Ethnicity MCQ By OpenStax Start Quiz
	29 Biology 29 Vertebrates MCQ By OpenStax Start Quiz
	Professional Writing MCQ By Mary Cohen Start Quiz
	Theater History Final - Review By Cameron Casey Start Exam
	Real Estate Finance By Mldelatte Start Quiz
	46 Biology 46 Ecosystems MCQ By OpenStax Start Quiz

6.4 Graphs of logarithmic functions (Page 3/8)

Graphing a logarithmic function with the form f ( x ) = log b ( x ).

Graphing transformations of logarithmic functions

Graphing a horizontal shift of f ( x ) = log b ( x )

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

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

Questions & Answers

Read also:

Get Jobilize Job Search Mobile App in your pocket Now!

Graphing a logarithmic function with the form f ( x ) = log _b ( x ).

Graphing a horizontal shift of f ( x ) = log _b ( x )

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

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