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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

what is math number
Tric Reply
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
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
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Adu
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
If f(x) = x-2 then, f(3) when 5f(x+1) 5((3-2)+1) 5(1+1) 5(2) 10
Augustine
how do they get the third part x = (32)5/4
kinnecy Reply
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
how
Sheref
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar

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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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