## Section6.2Vertical 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.

### Subsection6.2.1Effects 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.