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  4. Graphs of exponential functions

4.1 Graphs of exponential functions  (Page 5/6)

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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).

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.

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.

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

what does the ideal gas law states
Joy Reply
Three charges q_{1}=+3\mu C, q_{2}=+6\mu C and q_{3}=+8\mu C are located at (2,0)m (0,0)m and (0,3) coordinates respectively. Find the magnitude and direction acted upon q_{2} by the two other charges.Draw the correct graphical illustration of the problem above showing the direction of all forces.
Kate Reply
To solve this problem, we need to first find the net force acting on charge q_{2}. The magnitude of the force exerted by q_{1} on q_{2} is given by F=\frac{kq_{1}q_{2}}{r^{2}} where k is the Coulomb constant, q_{1} and q_{2} are the charges of the particles, and r is the distance between them.
Muhammed
What is the direction and net electric force on q_{1}= 5µC located at (0,4)r due to charges q_{2}=7mu located at (0,0)m and q_{3}=3\mu C located at (4,0)m?
Kate Reply
what is the change in momentum of a body?
Eunice Reply
what is a capacitor?
Raymond Reply
Capacitor is a separation of opposite charges using an insulator of very small dimension between them. Capacitor is used for allowing an AC (alternating current) to pass while a DC (direct current) is blocked.
Gautam
A motor travelling at 72km/m on sighting a stop sign applying the breaks such that under constant deaccelerate in the meters of 50 metres what is the magnitude of the accelerate
Maria Reply
please solve
Sharon
8m/s²
Aishat
What is Thermodynamics
Muordit
velocity can be 72 km/h in question. 72 km/h=20 m/s, v^2=2.a.x , 20^2=2.a.50, a=4 m/s^2.
Mehmet
A boat travels due east at a speed of 40meter per seconds across a river flowing due south at 30meter per seconds. what is the resultant speed of the boat
Saheed Reply
50 m/s due south east
Someone
which has a higher temperature, 1cup of boiling water or 1teapot of boiling water which can transfer more heat 1cup of boiling water or 1 teapot of boiling water explain your . answer
Ramon Reply
I believe temperature being an intensive property does not change for any amount of boiling water whereas heat being an extensive property changes with amount/size of the system.
Someone
Scratch that
Someone
temperature for any amount of water to boil at ntp is 100⁰C (it is a state function and and intensive property) and it depends both will give same amount of heat because the surface available for heat transfer is greater in case of the kettle as well as the heat stored in it but if you talk.....
Someone
about the amount of heat stored in the system then in that case since the mass of water in the kettle is greater so more energy is required to raise the temperature b/c more molecules of water are present in the kettle
Someone
definitely of physics
Haryormhidey Reply
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Esrael Reply
what is field
Felix Reply
physics, biology and chemistry this is my Field
ALIYU
field is a region of space under the influence of some physical properties
Collete
what is ogarnic chemistry
WISDOM Reply
determine the slope giving that 3y+ 2x-14=0
WISDOM
Another formula for Acceleration
Belty Reply
a=v/t. a=f/m a
IHUMA
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Adah
pratica A on solution of hydro chloric acid,B is a solution containing 0.5000 mole ofsodium chlorid per dm³,put A in the burret and titrate 20.00 or 25.00cm³ portion of B using melting orange as the indicator. record the deside of your burret tabulate the burret reading and calculate the average volume of acid used?
Nassze Reply
how do lnternal energy measures
Esrael
Two bodies attract each other electrically. Do they both have to be charged? Answer the same question if the bodies repel one another.
JALLAH Reply
No. According to Isac Newtons law. this two bodies maybe you and the wall beside you. Attracting depends on the mass och each body and distance between them.
Dlovan
Are you really asking if two bodies have to be charged to be influenced by Coulombs Law?
Robert
like charges repel while unlike charges atttact
Raymond
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Source:  OpenStax, Essential precalculus, part 1. OpenStax CNX. Aug 26, 2015 Download for free at http://legacy.cnx.org/content/col11871/1.1
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