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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

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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
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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.
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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 .

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Questions & Answers

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
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Robert
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Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
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Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
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meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
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meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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