# 3.5 Transformation of functions  (Page 8/21)

 Page 8 / 21

Write the formula for the function that we get when we stretch the identity toolkit function by a factor of 3, and then shift it down by 2 units.

$g\left(x\right)=3x-2$

## Horizontal stretches and compressions

Now we consider changes to the inside of a function. When we multiply a function’s input by a positive constant, we get a function whose graph is stretched or compressed horizontally in relation to the graph of the original function. If the constant is between 0 and 1, we get a horizontal stretch ; if the constant is greater than 1, we get a horizontal compression of the function.

Given a function $\text{\hspace{0.17em}}y=f\left(x\right),\text{\hspace{0.17em}}$ the form $\text{\hspace{0.17em}}y=f\left(bx\right)\text{\hspace{0.17em}}$ results in a horizontal stretch or compression. Consider the function $\text{\hspace{0.17em}}y={x}^{2}.\text{\hspace{0.17em}}$ Observe [link] . The graph of $\text{\hspace{0.17em}}y={\left(0.5x\right)}^{2}\text{\hspace{0.17em}}$ is a horizontal stretch of the graph of the function $\text{\hspace{0.17em}}y={x}^{2}\text{\hspace{0.17em}}$ by a factor of 2. The graph of $\text{\hspace{0.17em}}y={\left(2x\right)}^{2}\text{\hspace{0.17em}}$ is a horizontal compression of the graph of the function $\text{\hspace{0.17em}}y={x}^{2}\text{\hspace{0.17em}}$ by a factor of 2.

## Horizontal stretches and compressions

Given a function $\text{\hspace{0.17em}}f\left(x\right),\text{\hspace{0.17em}}$ a new function $\text{\hspace{0.17em}}g\left(x\right)=f\left(bx\right),\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}b\text{\hspace{0.17em}}$ is a constant, is a horizontal stretch    or horizontal compression    of the function $\text{\hspace{0.17em}}f\left(x\right).$

• If $\text{\hspace{0.17em}}b>1,\text{\hspace{0.17em}}$ then the graph will be compressed by $\text{\hspace{0.17em}}\frac{1}{b}.$
• If $\text{\hspace{0.17em}}0 then the graph will be stretched by $\text{\hspace{0.17em}}\frac{1}{b}.$
• If $\text{\hspace{0.17em}}b<0,\text{\hspace{0.17em}}$ then there will be combination of a horizontal stretch or compression with a horizontal reflection.

Given a description of a function, sketch a horizontal compression or stretch.

1. Write a formula to represent the function.
2. Set $\text{\hspace{0.17em}}g\left(x\right)=f\left(bx\right)\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}b>1\text{\hspace{0.17em}}$ for a compression or $\text{\hspace{0.17em}}0 for a stretch.

## Graphing a horizontal compression

Suppose a scientist is comparing a population of fruit flies to a population that progresses through its lifespan twice as fast as the original population. In other words, this new population, $\text{\hspace{0.17em}}R,\text{\hspace{0.17em}}$ will progress in 1 hour the same amount as the original population does in 2 hours, and in 2 hours, it will progress as much as the original population does in 4 hours. Sketch a graph of this population.

Symbolically, we could write

See [link] for a graphical comparison of the original population and the compressed population.

## Finding a horizontal stretch for a tabular function

A function $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ is given as [link] . Create a table for the function $\text{\hspace{0.17em}}g\left(x\right)=f\left(\frac{1}{2}x\right).$

 $x$ 2 4 6 8 $f\left(x\right)$ 1 3 7 11

The formula $\text{\hspace{0.17em}}g\left(x\right)=f\left(\frac{1}{2}x\right)\text{\hspace{0.17em}}$ tells us that the output values for $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ are the same as the output values for the function $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ at an input half the size. Notice that we do not have enough information to determine $\text{\hspace{0.17em}}g\left(2\right)\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}g\left(2\right)=f\left(\frac{1}{2}\cdot 2\right)=f\left(1\right),\text{\hspace{0.17em}}$ and we do not have a value for $\text{\hspace{0.17em}}f\left(1\right)\text{\hspace{0.17em}}$ in our table. Our input values to $\text{\hspace{0.17em}}g\text{\hspace{0.17em}}$ will need to be twice as large to get inputs for $\text{\hspace{0.17em}}f\text{\hspace{0.17em}}$ that we can evaluate. For example, we can determine $\text{\hspace{0.17em}}g\left(4\right)\text{.}$

$g\left(4\right)=f\left(\frac{1}{2}\cdot 4\right)=f\left(2\right)=1$

We do the same for the other values to produce [link] .

 $x$ 4 8 12 16 $g\left(x\right)$ 1 3 7 11

[link] shows the graphs of both of these sets of points.

## Recognizing a horizontal compression on a graph

Relate the function $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ to $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ in [link] .

The graph of $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ looks like the graph of $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ horizontally compressed. Because $\text{\hspace{0.17em}}f\left(x\right)\text{\hspace{0.17em}}$ ends at $\text{\hspace{0.17em}}\left(6,4\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(x\right)\text{\hspace{0.17em}}$ ends at $\text{\hspace{0.17em}}\left(2,4\right),\text{\hspace{0.17em}}$ we can see that the $\text{\hspace{0.17em}}x\text{-}$ values have been compressed by $\text{\hspace{0.17em}}\frac{1}{3},\text{\hspace{0.17em}}$ because $\text{\hspace{0.17em}}6\left(\frac{1}{3}\right)=2.\text{\hspace{0.17em}}$ We might also notice that $\text{\hspace{0.17em}}g\left(2\right)=f\left(6\right)\text{\hspace{0.17em}}$ and $\text{\hspace{0.17em}}g\left(1\right)=f\left(3\right).\text{\hspace{0.17em}}$ Either way, we can describe this relationship as $\text{\hspace{0.17em}}g\left(x\right)=f\left(3x\right).\text{\hspace{0.17em}}$ This is a horizontal compression by $\text{\hspace{0.17em}}\frac{1}{3}.$

#### 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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