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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

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.
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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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