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Section 6.2 Vertical Stretches

In this activity, we will continue to explore how a change to a function's formula will alter its graph. In particular, we will see the effect of multiplying the output of a function by a constant number:

\begin{equation*} k\cdot f(x) \end{equation*}

Remember that if you perform some calculation with the output of a function \(f\text{,}\) you will be performing that calculation with an expression like \(f(a)\text{.}\)

In this first exercise, you will practice writing an expression for the output of a function which has been altered.

In the next exercise, you will practice interpreting function notation where the input has been altered.

Now, remembering that multiplying outside a function will alter the outputs of the function, interpret the meaning vertical stretches in the next exercise.

Subsection 6.2.1 Effects on the graph

Now we will see how to apply these concepts to the graph of a function.

First, you will use the graph of a function to complete a table of values by altering the output.

Next, you will plot points on the graphs of functions where the output has been multiplied by a number.

In the previous exercise, you plotted points for the functions \(y = 2f(x)\) and \(\frac{1}{2}f(x)\text{.}\) This created graphs which were similar to the original function \(y = f(x)\text{,}\) but whose outputs were either twice as large, or half as large as those of \(f(x)\text{.}\)

This makes sense, because multiplying outside of the function \(f(x)\) by a number like \(2\) or \(\frac{1}{2}\) only changes the output values — and the output values are the heights of the points.

In the next two exercises, you will continue to explore how multiplying the output of a function by a number will change the graph.

One use of stretching or compressing a function vertically is to make it match a given point on the graph.

Finally, compare the transformation \(a\cdot f(x)\) with the transformation \(f(x) + a\) which we studied in chapter 5.