Graphing Functions Using Stretches and Compressions
Adding a constant to the inputs or outputs of a function changed the position of a graph with respect to the axes, but it did not affect the shape of a graph. We now explore the effects of multiplying the inputs or outputs by some quantity.
Vertical Stretches and Compressions
We can transform the inside (input values) of a function or we can transform the outside (output values) of a function. Each change has a specific effect that can be seen graphically.
When we multiply a function by a positive constant, we get a function whose graph is stretched or compressed vertically in relation to the graph of the original function. If the constant is greater than 1, we get a vertical stretch; if the constant is between 0 and 1, we get a vertical compression. The graph below shows a function multiplied by constant factors 2 and 0.5 and the resulting vertical stretch and compression.
Vertical stretch and compression
vertical stretches and compressions
A vertical stretch or compression involves scaling the graph of a function [latex]f(x)[/latex] by a constant factor [latex]a[/latex].
[latex]g(x) = a \cdot f(x)[/latex]
This transformation changes the output values of the function.
If [latex]a>1[/latex]: The graph is stretched vertically.
If [latex]0 < a < 1[/latex]: The graph is compressed vertically.
If [latex]a<0[/latex]: A combination of vertical stretch/compression and vertical reflection.
How To: Given a function, graph its vertical stretch.
Identify the value of [latex]a[/latex].
Multiply all range values by [latex]a[/latex].
If [latex]a>1[/latex], the graph is stretched by a factor of [latex]a[/latex].
If [latex]{ 0 }<{ a }<{ 1 }[/latex], the graph is compressed by a factor of [latex]a[/latex].
If [latex]a<0[/latex], the graph is either stretched or compressed and also reflected about the [latex]x[/latex]-axis.
A function [latex]P\left(t\right)[/latex] models the number of fruit flies in a population over time, and is graphed below.A scientist is comparing this population to another population, [latex]Q[/latex], whose growth follows the same pattern, but is twice as large. Sketch a graph of this population.
Graph to represent the growth of the population of fruit flies
Because the population is always twice as large, the new population’s output values are always twice the original function’s output values.If we choose four reference points, [latex](0, 1)[/latex], [latex](3, 3)[/latex], [latex](6, 2)[/latex] and [latex](7, 0)[/latex] we will multiply all of the outputs by [latex]2[/latex].The following shows where the new points for the new graph will be located.
This means that for any input [latex]t[/latex], the value of the function [latex]Q[/latex] is twice the value of the function [latex]P[/latex]. 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, [latex]t[/latex], stay the same while the output values are twice as large as before.
A function [latex]f[/latex] is given in the table below. Create a table for the function [latex]g\left(x\right)=\frac{1}{2}f\left(x\right)[/latex].
[latex]x[/latex]
[latex]2[/latex]
[latex]4[/latex]
[latex]6[/latex]
[latex]8[/latex]
[latex]f\left(x\right)[/latex]
[latex]1[/latex]
[latex]3[/latex]
[latex]7[/latex]
[latex]11[/latex]
The formula [latex]g\left(x\right)=\frac{1}{2}f\left(x\right)[/latex] tells us that the output values of [latex]g[/latex] are half of the output values of [latex]f[/latex] with the same inputs. For example, we know that [latex]f\left(4\right)=3[/latex]. Then:
We do the same for the other values to produce this table.
[latex]x[/latex]
[latex]2[/latex]
[latex]4[/latex]
[latex]6[/latex]
[latex]8[/latex]
[latex]g\left(x\right)[/latex]
[latex]\frac{1}{2}[/latex]
[latex]\frac{3}{2}[/latex]
[latex]\frac{7}{2}[/latex]
[latex]\frac{11}{2}[/latex]
[latex]\\[/latex] Analysis of the Solution
The result is that the function [latex]g\left(x\right)[/latex] has been compressed vertically by [latex]\frac{1}{2}[/latex]. Each output value is divided in half, so the graph is half the original height.
Graph of g(x) and f(x)
The graph shows two function: The toolkit function [latex]f(x) = x^3[/latex] (green) and [latex]g(x)[/latex] (red).Relate this new function [latex]g\left(x\right)[/latex] to [latex]f\left(x\right)[/latex], and then find a formula for [latex]g\left(x\right)[/latex].
The red curve [latex]g(x)[/latex] appears to be less steep compared to the green curve [latex]f(x)[/latex]. This suggests a vertical compression.If [latex]g(x)[/latex] is a vertical compression of [latex]f(x)[/latex], we have: [latex]g(x) = a \cdot f(x)[/latex], where [latex]0 < a < 1[/latex].To determine [latex]a[/latex], it is helpful to look for a point on the graph that is relatively clear.
In this graph, it appears that [latex]g\left(2\right)=2[/latex].
With the basic cubic function at the same input, [latex]f\left(2\right)={2}^{3}=8[/latex].
Based on that, it appears that the outputs of [latex]g[/latex] are [latex]\frac{1}{4}[/latex] the outputs of the function [latex]f[/latex] because [latex]2=\frac{1}{4} \cdot 8[/latex].
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.
Given a function [latex]y=f\left(x\right)[/latex], the form [latex]y=f\left(bx\right)[/latex] results in a horizontal stretch or compression. Consider the function [latex]y={x}^{2}[/latex]. The graph of [latex]y={\left(0.5x\right)}^{2}[/latex] is a horizontal stretch of the graph of the function [latex]y={x}^{2}[/latex] by a factor of 2. The graph of [latex]y={\left(2x\right)}^{2}[/latex] is a horizontal compression of the graph of the function [latex]y={x}^{2}[/latex] by a factor of [latex]2[/latex].
Graph of the vertical stretch and compression of x^2
horizontal stretches and compressions
A horizontal stretch or compression involves scaling the graph of a function [latex]f(x)[/latex] by a constant factor [latex]b[/latex].
[latex]g(x) = f(b \cdot x)[/latex]
This transformation changes the input values of the function.
If [latex]b>1[/latex]: The graph is compressed horizontally. The graph is compressed by [latex]\dfrac{1}{b}[/latex].
If [latex]0 < b < 1[/latex]: The graph is stretched horizontally. The graph is stretched by [latex]\dfrac{1}{b}[/latex].
If [latex]b<0[/latex]: A combination of horizontal stretch/compression and horizontal reflection.
How To: Given a description of a function, sketch a horizontal compression or stretch.
Write a formula to represent the function.
Set [latex]g\left(x\right)=f\left(bx\right)[/latex] where [latex]b{\gt}1[/latex] for a compression or [latex]0{\lt}b{\lt}1[/latex] for a stretch.
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, [latex]R[/latex], will progress in [latex]1[/latex] hour the same amount as the original population does in [latex]2[/latex] hours, and in [latex]2[/latex] hours, it will progress as much as the original population does in [latex]4[/latex] hours. Sketch a graph of this population.
Symbolically, we could write
[latex]\begin{align}&R\left(1\right)=P\left(2\right), \\ &R\left(2\right)=P\left(4\right),\text{ and in general,} \\ &R\left(t\right)=P\left(2t\right). \end{align}[/latex]
See below for a graphical comparison of the original population and the compressed population.
(a) Original population graph (b) Compressed population graph
[latex]\\[/latex] Analysis of the Solution
[latex]\\[/latex]
Note that the effect on the graph is a horizontal compression where all input values are half of their original distance from the vertical axis.
A function [latex]f\left(x\right)[/latex] is given below. Create a table for the function [latex]g\left(x\right)=f\left(\frac{1}{2}x\right)[/latex].
[latex]x[/latex]
[latex]2[/latex]
[latex]4[/latex]
[latex]6[/latex]
[latex]8[/latex]
[latex]f\left(x\right)[/latex]
[latex]1[/latex]
[latex]3[/latex]
[latex]7[/latex]
[latex]11[/latex]
The formula [latex]g\left(x\right)=f\left(\frac{1}{2}x\right)[/latex] tells us that the output values for [latex]g[/latex] are the same as the output values for the function [latex]f[/latex] at an input half the size. Notice that we do not have enough information to determine [latex]g\left(2\right)[/latex] because [latex]g\left(2\right)=f\left(\frac{1}{2}\cdot 2\right)=f\left(1\right)[/latex], and we do not have a value for [latex]f\left(1\right)[/latex] in our table. Our input values to [latex]g[/latex] will need to be twice as large to get inputs for [latex]f[/latex] that we can evaluate. For example, we can determine [latex]g\left(4\right)\text{.}[/latex]
We do the same for the other values to produce the table below.
[latex]x[/latex]
4
8
12
16
[latex]g\left(x\right)[/latex]
1
3
7
11
Graph of the previous table
This figure shows the graphs of both of these sets of points.
Analysis of the Solution
Because each input value has been doubled, the result is that the function [latex]g\left(x\right)[/latex] has been stretched horizontally by a factor of 2.
Graph of f(x) being vertically compressed to g(x)
Relate the function [latex]g\left(x\right)[/latex] to [latex]f\left(x\right)[/latex].
The orange graph [latex]g(x)[/latex] appears to be a horizontally compressed version of the blue graph of [latex]f(x)[/latex].
[latex]\\[/latex]
If [latex]g(x)[/latex] is a horizontal compression of [latex]f(x)[/latex], we have: [latex]g(x) = f(b \cdot x)[/latex], where [latex]b > 1[/latex]. The graph is compressed by [latex]\dfrac{1}{b}[/latex].To determine [latex]b[/latex], it is helpful to look for a point on the graph that is relatively clear.
In the compressed graph [latex]g(x)[/latex], the end point is [latex](2, 4)[/latex].
The end point of [latex]f(x)[/latex] is [latex](6,4)[/latex].
We can see that the [latex]x[/latex]-values have been compressed by [latex]\frac{1}{3}[/latex], because [latex]2=\frac{1}{3} \cdot 6[/latex].
This means that [latex]\dfrac{1}{b} = \dfrac{1}{3}[/latex], which means [latex]b = 3[/latex].
![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).](https://s3-us-west-2.amazonaws.com/courses-images/wp-content/uploads/sites/896/2016/10/18203623/CNX_Precalc_Figure_01_05_029ab.jpg)
