1. Home
  2. College algebra
  3. Exponential and logarithmic
  4. Graphs of exponential functions

6.2 Graphs of exponential functions  (Page 5/6)

<< Chapter < Page
  College algebra     Page 5 / 6
Chapter >> Page >

Find and graph the equation for a function,   g ( x ) , that reflects   f ( x ) = 1.25 x   about the y -axis. State its domain, range, and asymptote.

The domain is   ( − ∞ , ∞ ) ;   the range is   ( 0 , ∞ ) ;   the horizontal asymptote is   y = 0.

Graph of the function, g(x) = -(1.25)^(-x), with an asymptote at y=0. Labeled points in the graph are (-1, 1.25), (0, 1), and (1, 0.8).
Got questions? Get instant answers now!

Summarizing translations of the exponential function

Now that we have worked with each type of translation for the exponential function, we can summarize them in [link] to arrive at the general equation for translating exponential functions.

Translations of the Parent Function   f ( x ) = b x
Translation Form
Shift
  • Horizontally   c   units to the left
  • Vertically   d   units up
 f ( x ) = b x + c + d
Stretch and Compress
  • Stretch if   | a | > 1
  • Compression if   0 < | a | < 1
f ( x ) = a b x
Reflect about the x -axis f ( x ) = − b x
Reflect about the y -axis f ( x ) = b − x = ( 1 b ) x
General equation for all translations f ( x ) = a b x + c + d

Translations of exponential functions

A translation of an exponential function has the form

  f ( x ) = a b x + c + d

Where the parent function,   y = b x ,   b > 1 , is

  • shifted horizontally   c   units to the left.
  • stretched vertically by a factor of   | a |   if   | a | > 0.
  • compressed vertically by a factor of   | a |   if   0 < | a | < 1.
  • shifted vertically   d   units.
  • reflected about the x- axis when   a < 0.

Note the order of the shifts, transformations, and reflections follow the order of operations.

Writing a function from a description

Write the equation for the function described below. Give the horizontal asymptote, the domain, and the range.

  • f ( x ) = e x   is vertically stretched by a factor of   2   , reflected across the y -axis, and then shifted up   4   units.

We want to find an equation of the general form     f ( x ) = a b x + c + d .   We use the description provided to find   a , b , c , and   d .

  • We are given the parent function   f ( x ) = e x , so   b = e .
  • The function is stretched by a factor of   2 , so   a = 2.
  • The function is reflected about the y -axis. We replace   x   with   − x   to get:   e − x .
  • The graph is shifted vertically 4 units, so   d = 4.

Substituting in the general form we get,

  f ( x ) = a b x + c + d = 2 e − x + 0 + 4 = 2 e − x + 4

The domain is   ( − ∞ , ∞ ) ;   the range is   ( 4 , ∞ ) ;   the horizontal asymptote is   y = 4.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Write the equation for function described below. Give the horizontal asymptote, the domain, and the range.

  • f ( x ) = e x   is compressed vertically by a factor of   1 3 , reflected across the x -axis and then shifted down   2 units.

f ( x ) = − 1 3 e x − 2 ;   the domain is   ( − ∞ , ∞ ) ;   the range is   ( − ∞ , 2 ) ;   the horizontal asymptote is   y = 2.

Got questions? Get instant answers now!

Access this online resource for additional instruction and practice with graphing exponential functions.

Key equations

General Form for the Translation of the Parent Function   f ( x ) = b x f ( x ) = a b x + c + d

Key concepts

  • The graph of the function   f ( x ) = b x   has a y- intercept at   ( 0 ,   1 ) , domain   ( − ∞ ,   ∞ ) , range   ( 0 ,   ∞ ) , and horizontal asymptote   y = 0.   See [link] .
  • If   b > 1 , the function is increasing. The left tail of the graph will approach the asymptote   y = 0 , and the right tail will increase without bound.
  • If   0 < b < 1 , the function is decreasing. The left tail of the graph will increase without bound, and the right tail will approach the asymptote   y = 0.
  • The equation   f ( x ) = b x + d   represents a vertical shift of the parent function   f ( x ) = b x .
  • The equation   f ( x ) = b x + c   represents a horizontal shift of the parent function   f ( x ) = b x .   See [link] .
  • Approximate solutions of the equation   f ( x ) = b x + c + d   can be found using a graphing calculator. See [link] .
  • The equation   f ( x ) = a b x , where   a > 0 , represents a vertical stretch if   | a | > 1   or compression if   0 < | a | < 1   of the parent function   f ( x ) = b x .   See [link] .
  • When the parent function   f ( x ) = b x   is multiplied by   − 1 , the result,   f ( x ) = − b x , is a reflection about the x -axis. When the input is multiplied by   − 1 , the result,   f ( x ) = b − x , is a reflection about the y -axis. See [link] .
  • All translations of the exponential function can be summarized by the general equation   f ( x ) = a b x + c + d .   See [link] .
  • Using the general equation   f ( x ) = a b x + c + d , we can write the equation of a function given its description. See [link] .

Questions & Answers

how do you get the 2/50
Abba Reply
number of sport play by 50 student construct discrete data
Aminu Reply
width of the frangebany leaves on how to write a introduction
Theresa Reply
Solve the mean of variance
Veronica Reply
Step 1: Find the mean. To find the mean, add up all the scores, then divide them by the number of scores. ... Step 2: Find each score's deviation from the mean. ... Step 3: Square each deviation from the mean. ... Step 4: Find the sum of squares. ... Step 5: Divide the sum of squares by n – 1 or N.
kenneth
what is error
Yakuba Reply
Is mistake done to something
Vutshila
Hy
anas
hy
What is the life teble
anas
hy
Jibrin
statistics is the analyzing of data
Tajudeen Reply
what is statics?
Zelalem Reply
how do you calculate mean
Gloria Reply
diveving the sum if all values
Shaynaynay
let A1,A2 and A3 events be independent,show that (A1)^c, (A2)^c and (A3)^c are independent?
Fisaye Reply
what is statistics
Akhisani Reply
data collected all over the world
Shaynaynay
construct a less than and more than table
Imad Reply
The sample of 16 students is taken. The average age in the sample was 22 years with astandard deviation of 6 years. Construct a 95% confidence interval for the age of the population.
Aschalew Reply
Bhartdarshan' is an internet-based travel agency wherein customer can see videos of the cities they plant to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400 a. what is the probability of getting more than 12,000 hits? b. what is the probability of getting fewer than 9,000 hits?
Akshay Reply
Bhartdarshan'is an internet-based travel agency wherein customer can see videos of the cities they plan to visit. The number of hits daily is a normally distributed random variable with a mean of 10,000 and a standard deviation of 2,400. a. What is the probability of getting more than 12,000 hits
Akshay
1
Bright
Sorry i want to learn more about this question
Bright
Someone help
Bright
a= 0.20233 b=0.3384
Sufiyan
a
Shaynaynay
How do I interpret level of significance?
Mohd Reply
It depends on your business problem or in Machine Learning you could use ROC- AUC cruve to decide the threshold value
Shivam
how skewness and kurtosis are used in statistics
Owen Reply
yes what is it
Taneeya
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!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'College algebra' conversation and receive update notifications?

Ask