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

find the equation of the tangent to the curve y=2x³-x²+3x+1 at the points x=1 and x=3
Esther Reply
derivative of logarithms function
Iqra Reply
how to solve this question
sidra
ex 2.1 question no 11
khansa
anyone can help me
khansa
question please
Rasul
ex 2.1 question no. 11
khansa
i cant type here
khansa
Find the derivative of g(x)=−3.
Abdullah Reply
any genius online ? I need help!!
Guzorochi Reply
how can i help you?
Pina
need to learn polynomial
Zakariya
i will teach...
nandu
I'm waiting
Zakariya
plz help me in question
Abish
How can I help you?
Tlou
evaluate the following computation (x³-8/x-2)
Murtala Reply
teach me how to solve the first law of calculus.
Uncle Reply
teach me also how to solve the first law of calculus
Bilson
what is differentiation
Ibrahim Reply
only god knows😂
abdulkadir
f(x) = x-2 g(x) = 3x + 5 fog(x)? f(x)/g(x)
Naufal Reply
fog(x)= f(g(x)) = x-2 = 3x+5-2 = 3x+3 f(x)/g(x)= x-2/3x+5
diron
pweding paturo nsa calculus?
jimmy
how to use fundamental theorem to solve exponential
JULIA Reply
find the bounded area of the parabola y^2=4x and y=16x
Omar Reply
what is absolute value means?
Geo Reply
Chicken nuggets
Hugh
🐔
MM
🐔🦃 nuggets
MM
(mathematics) For a complex number a+bi, the principal square root of the sum of the squares of its real and imaginary parts, √a2+b2 . Denoted by | |. The absolute value |x| of a real number x is √x2 , which is equal to x if x is non-negative, and −x if x is negative.
Ismael
find integration of loge x
Game Reply
find the volume of a solid about the y-axis, x=0, x=1, y=0, y=7+x^3
Godwin Reply
how does this work
Brad Reply
Can calculus give the answers as same as other methods give in basic classes while solving the numericals?
Cosmos Reply
log tan (x/4+x/2)
Rohan
please answer
Rohan
y=(x^2 + 3x).(eipix)
Claudia
is this a answer
Ismael
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