<< Chapter < Page Chapter >> Page >

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 ( x ) = 3 x - 2

Got questions? Get instant answers now!

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.

Graph of the vertical stretch and compression of x^2.

Given a function y = f ( x ) , the form y = f ( b x ) results in a horizontal stretch or compression. Consider the function y = x 2 . Observe [link] . The graph of y = ( 0.5 x ) 2 is a horizontal stretch of the graph of the function y = x 2 by a factor of 2. The graph of y = ( 2 x ) 2 is a horizontal compression of the graph of the function y = x 2 by a factor of 2.

Horizontal stretches and compressions

Given a function f ( x ) , a new function g ( x ) = f ( b x ) , where b is a constant, is a horizontal stretch    or horizontal compression    of the function f ( x ) .

  • If b > 1 , then the graph will be compressed by 1 b .
  • If 0 < b < 1 , then the graph will be stretched by 1 b .
  • If b < 0 , 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 g ( x ) = f ( b x ) where b > 1 for a compression or 0 < b < 1 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, R , 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

R ( 1 ) = P ( 2 ) , R ( 2 ) = P ( 4 ) ,  and in general, R ( t ) = P ( 2 t ) .

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

Two side-by-side graphs. The first graph has function for original population whose domain is [0,7] and range is [0,3]. The maximum value occurs at (3,3). The second graph has the same shape as the first except it is half as wide. It is a graph of transformed population, with a domain of [0, 3.5] and a range of [0,3]. The maximum occurs at (1.5, 3).
(a) Original population graph (b) Compressed population graph
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Finding a horizontal stretch for a tabular function

A function f ( x ) is given as [link] . Create a table for the function g ( x ) = f ( 1 2 x ) .

x 2 4 6 8
f ( x ) 1 3 7 11

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

g ( 4 ) = f ( 1 2 4 ) = f ( 2 ) = 1

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

x 4 8 12 16
g ( x ) 1 3 7 11

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

Graph of the previous table.
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Recognizing a horizontal compression on a graph

Relate the function g ( x ) to f ( x ) in [link] .

Graph of f(x) being vertically compressed to g(x).

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

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

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"
Vedant
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
SANDESH Reply
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
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
Give me the reciprocal of even number
Aliyu
The reciprocal of an even number is a proper fraction
Jamilu
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc 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