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Two graphs where graph a is an example of vertical stretch and graph b is an example of vertical compression.
(a) g ( x ) = 3 ( 2 ) x stretches the graph of f ( x ) = 2 x vertically by a factor of 3. (b) h ( x ) = 1 3 ( 2 ) x compresses the graph of f ( x ) = 2 x vertically by a factor of 1 3 .

Stretches and compressions of the parent function f ( x ) = b x

For any factor a > 0 , the function f ( x ) = a ( b ) x

  • is stretched vertically by a factor of a if | a | > 1.
  • is compressed vertically by a factor of a if | a | < 1.
  • has a y -intercept of ( 0 , a ) .
  • has a horizontal asymptote at y = 0 , a range of ( 0 , ) , and a domain of ( , ) , which are unchanged from the parent function.

Graphing the stretch of an exponential function

Sketch a graph of f ( x ) = 4 ( 1 2 ) x . State the domain, range, and asymptote.

Before graphing, identify the behavior and key points on the graph.

  • Since b = 1 2 is between zero and one, the left tail of the graph will increase without bound as x decreases, and the right tail will approach the x -axis as x increases.
  • Since a = 4 , the graph of f ( x ) = ( 1 2 ) x will be stretched by a factor of 4.
  • Create a table of points as shown in [link] .
    x 3 2 1 0 1 2 3
    f ( x ) = 4 ( 1 2 ) x 32 16 8 4 2 1 0.5
  • Plot the y- intercept, ( 0 , 4 ) , along with two other points. We can use ( 1 , 8 ) and ( 1 , 2 ) .

Draw a smooth curve connecting the points, as shown in [link] .

Graph of the function, f(x) = 4(1/2)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 8), (0, 4), and (1, 2).

The domain is ( , ) ; the range is ( 0 , ) ; the horizontal asymptote is y = 0.

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Sketch the graph of f ( x ) = 1 2 ( 4 ) x . State the domain, range, and asymptote.

The domain is ( , ) ; the range is ( 0 , ) ; the horizontal asymptote is y = 0.
Graph of the function, f(x) = (1/2)(4)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, 0.125), (0, 0.5), and (1, 2).

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Graphing reflections

In addition to shifting, compressing, and stretching a graph, we can also reflect it about the x -axis or the y -axis. When we multiply the parent function f ( x ) = b x by −1 , we get a reflection about the x -axis. When we multiply the input by −1 , we get a reflection about the y -axis. For example, if we begin by graphing the parent function f ( x ) = 2 x , we can then graph the two reflections alongside it. The reflection about the x -axis, g ( x ) = −2 x , is shown on the left side of [link] , and the reflection about the y -axis h ( x ) = 2 x , is shown on the right side of [link] .

Two graphs where graph a is an example of a reflection about the x-axis and graph b is an example of a reflection about the y-axis.
(a) g ( x ) = 2 x reflects the graph of f ( x ) = 2 x about the x-axis. (b) g ( x ) = 2 x reflects the graph of f ( x ) = 2 x about the y -axis.

Reflections of the parent function f ( x ) = b x

The function f ( x ) = b x

  • reflects the parent function f ( x ) = b x about the x -axis.
  • has a y -intercept of ( 0 , 1 ) .
  • has a range of ( , 0 )
  • has a horizontal asymptote at y = 0 and domain of ( , ) , which are unchanged from the parent function.

The function f ( x ) = b x

  • reflects the parent function f ( x ) = b x about the y -axis.
  • has a y -intercept of ( 0 , 1 ) , a horizontal asymptote at y = 0 , a range of ( 0 , ) , and a domain of ( , ) , which are unchanged from the parent function.

Writing and graphing the reflection of an exponential function

Find and graph the equation for a function, g ( x ) , that reflects f ( x ) = ( 1 4 ) x about the x -axis. State its domain, range, and asymptote.

Since we want to reflect the parent function f ( x ) = ( 1 4 ) x about the x- axis, we multiply f ( x ) by 1 to get, g ( x ) = ( 1 4 ) x . Next we create a table of points as in [link] .

x 3 2 1 0 1 2 3
g ( x ) = ( 1 4 ) x 64 16 4 1 0.25 0.0625 0.0156

Plot the y- intercept, ( 0 , −1 ) , along with two other points. We can use ( −1 , −4 ) and ( 1 , −0.25 ) .

Draw a smooth curve connecting the points:

Graph of the function, g(x) = -(0.25)^(x), with an asymptote at y=0. Labeled points in the graph are (-1, -4), (0, -1), and (1, -0.25).

The domain is ( , ) ; the range is ( , 0 ) ; the horizontal asymptote is y = 0.

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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
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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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