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Graph of a function that shows vertical stretching and compression.
Vertical stretch and compression

Vertical stretches and compressions

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

  • If a > 1 , then the graph will be stretched.
  • If 0 < a < 1 , then the graph will be compressed.
  • If a < 0 , then there will be combination of a vertical stretch or compression with a vertical reflection.

Given a function, graph its vertical stretch.

  1. Identify the value of a .
  2. Multiply all range values by a .
  3. If a > 1 , the graph is stretched by a factor of a .

    If 0 < a < 1 , the graph is compressed by a factor of a .

    If a < 0 , the graph is either stretched or compressed and also reflected about the x -axis.

Graphing a vertical stretch

A function P ( t ) models the population of fruit flies. The graph is shown in [link] .

Graph to represent the growth of the population of fruit flies.

A scientist is comparing this population to another population, Q , whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.

Because the population is always twice as large, the new population’s output values are always twice the original function’s output values. Graphically, this is shown in [link] .

If we choose four reference points, (0, 1), (3, 3), (6, 2) and (7, 0) we will multiply all of the outputs by 2.

The following shows where the new points for the new graph will be located.

( 0 ,   1 ) ( 0 ,   2 ) ( 3 ,   3 ) ( 3 ,   6 ) ( 6 ,   2 ) ( 6 ,   4 ) ( 7 ,   0 ) ( 7 ,   0 )
Graph of the population function doubled.

Symbolically, the relationship is written as

Q ( t ) = 2 P ( t )

This means that for any input t , the value of the function Q is twice the value of the function P . Notice that the effect on the graph is a vertical stretching of the graph, where every point doubles its distance from the horizontal axis. The input values, t , stay the same while the output values are twice as large as before.

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Given a tabular function and assuming that the transformation is a vertical stretch or compression, create a table for a vertical compression.

  1. Determine the value of a .
  2. Multiply all of the output values by a .

Finding a vertical compression of a tabular function

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

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

The formula g ( x ) = 1 2 f ( x ) tells us that the output values of g are half of the output values of f with the same inputs. For example, we know that f ( 4 ) = 3. Then

g ( 4 ) = 1 2 f ( 4 ) = 1 2 ( 3 ) = 3 2

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

x 2 4 6 8
g ( x ) 1 2 3 2 7 2 11 2
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A function f is given as [link] . Create a table for the function g ( x ) = 3 4 f ( x ) .

x 2 4 6 8
f ( x ) 12 16 20 0
x 2 4 6 8
g ( x ) 9 12 15 0
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Recognizing a vertical stretch

The graph in [link] is a transformation of the toolkit function f ( x ) = x 3 . Relate this new function g ( x ) to f ( x ) , and then find a formula for g ( x ) .

Graph of a transformation of f(x)=x^3.

When trying to determine a vertical stretch or shift, it is helpful to look for a point on the graph that is relatively clear. In this graph, it appears that g ( 2 ) = 2. With the basic cubic function at the same input, f ( 2 ) = 2 3 = 8. Based on that, it appears that the outputs of g are 1 4 the outputs of the function f because g ( 2 ) = 1 4 f ( 2 ) . From this we can fairly safely conclude that g ( x ) = 1 4 f ( x ) .

We can write a formula for g by using the definition of the function f .

g ( x ) = 1 4 f ( x ) = 1 4 x 3
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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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