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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).
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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.

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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.

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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

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
hi mam
Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
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Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
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Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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