1. Home
  2. Algebra and trigonometry
  3. Exponential and logarithmic
  4. Graphs of exponential functions

6.2 Graphs of exponential functions  (Page 5/6)

<< Chapter < Page
  Algebra and trigonometry     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 to study physic and understand
Ewa Reply
what is conservative force with examples
Moses
what is work
Fredrick Reply
the transfer of energy by a force that causes an object to be displaced; the product of the component of the force in the direction of the displacement and the magnitude of the displacement
AI-Robot
why is it from light to gravity
Esther Reply
difference between model and theory
Esther
Is the ship moving at a constant velocity?
Kamogelo Reply
The full note of modern physics
aluet Reply
introduction to applications of nuclear physics
aluet Reply
the explanation is not in full details
Moses Reply
I need more explanation or all about kinematics
Moses
yes
zephaniah
I need more explanation or all about nuclear physics
aluet
Show that the equal masses particles emarge from collision at right angle by making explicit used of fact that momentum is a vector quantity
Muhammad Reply
yh
Isaac
A wave is described by the function D(x,t)=(1.6cm) sin[(1.2cm^-1(x+6.8cm/st] what are:a.Amplitude b. wavelength c. wave number d. frequency e. period f. velocity of speed.
Majok Reply
what is frontier of physics
Somto Reply
A body is projected upward at an angle 45° 18minutes with the horizontal with an initial speed of 40km per second. In hoe many seconds will the body reach the ground then how far from the point of projection will it strike. At what angle will the horizontal will strike
Gufraan Reply
Suppose hydrogen and oxygen are diffusing through air. A small amount of each is released simultaneously. How much time passes before the hydrogen is 1.00 s ahead of the oxygen? Such differences in arrival times are used as an analytical tool in gas chromatography.
Ezekiel Reply
please explain
Samuel
what's the definition of physics
Mobolaji Reply
what is physics
Nangun Reply
the science concerned with describing the interactions of energy, matter, space, and time; it is especially interested in what fundamental mechanisms underlie every phenomenon
AI-Robot
what is isotopes
Nangun Reply
nuclei having the same Z and different N s
AI-Robot
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, 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