<< Chapter < Page Chapter >> Page >

Write a formula for the toolkit square root function horizontally stretched by a factor of 3.

g ( x ) = f ( 1 3 x ) so using the square root function we get g ( x ) = 1 3 x

Got questions? Get instant answers now!

Performing a sequence of transformations

When combining transformations, it is very important to consider the order of the transformations. For example, vertically shifting by 3 and then vertically stretching by 2 does not create the same graph as vertically stretching by 2 and then vertically shifting by 3, because when we shift first, both the original function and the shift get stretched, while only the original function gets stretched when we stretch first.

When we see an expression such as 2 f ( x ) + 3 , which transformation should we start with? The answer here follows nicely from the order of operations. Given the output value of f ( x ) , we first multiply by 2, causing the vertical stretch, and then add 3, causing the vertical shift. In other words, multiplication before addition.

Horizontal transformations are a little trickier to think about. When we write g ( x ) = f ( 2 x + 3 ) , for example, we have to think about how the inputs to the function g relate to the inputs to the function f . Suppose we know f ( 7 ) = 12. What input to g would produce that output? In other words, what value of x will allow g ( x ) = f ( 2 x + 3 ) = 12 ? We would need 2 x + 3 = 7. To solve for x , we would first subtract 3, resulting in a horizontal shift, and then divide by 2, causing a horizontal compression.

This format ends up being very difficult to work with, because it is usually much easier to horizontally stretch a graph before shifting. We can work around this by factoring inside the function.

f ( b x + p ) = f ( b ( x + p b ) )

Let’s work through an example.

f ( x ) = ( 2 x + 4 ) 2

We can factor out a 2.

f ( x ) = ( 2 ( x + 2 ) ) 2

Now we can more clearly observe a horizontal shift to the left 2 units and a horizontal compression. Factoring in this way allows us to horizontally stretch first and then shift horizontally.

Combining transformations

When combining vertical transformations written in the form a f ( x ) + k , first vertically stretch by a and then vertically shift by k .

When combining horizontal transformations written in the form f ( b x + h ) , first horizontally shift by h and then horizontally stretch by 1 b .

When combining horizontal transformations written in the form f ( b ( x + h ) ) , first horizontally stretch by 1 b and then horizontally shift by h .

Horizontal and vertical transformations are independent. It does not matter whether horizontal or vertical transformations are performed first.

Finding a triple transformation of a tabular function

Given [link] for the function f ( x ) , create a table of values for the function g ( x ) = 2 f ( 3 x ) + 1.

x 6 12 18 24
f ( x ) 10 14 15 17

There are three steps to this transformation, and we will work from the inside out. Starting with the horizontal transformations, f ( 3 x ) is a horizontal compression by 1 3 , which means we multiply each x - value by 1 3 . See [link] .

x 2 4 6 8
f ( 3 x ) 10 14 15 17

Looking now to the vertical transformations, we start with the vertical stretch, which will multiply the output values by 2. We apply this to the previous transformation. See [link] .

x 2 4 6 8
2 f ( 3 x ) 20 28 30 34

Finally, we can apply the vertical shift, which will add 1 to all the output values. See [link] .

x 2 4 6 8
g ( x ) = 2 f ( 3 x ) + 1 21 29 31 35
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask