<< 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

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
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Please prove it
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
the 28th term is 175
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron 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?