## 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.

###### Exercise6.2.2

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

###### Exercise6.2.3

WeBWorK 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.

###### Exercise6.2.4

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

###### Exercise6.2.6

WeBWorK ExerciseIn 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.

###### Exercise6.2.7

WeBWorK Exercise###### Exercise6.2.8

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

###### Exercise6.2.9

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