In this activity, we will explore how to transform the graph of a function \(f\) by making changes to its formula in the following ways: \(f(x) + k\) and \(f(x + h)\text{.}\) By the end of this activity, you will be able to:

Recognize the geometric effect of transforming \(f(x)\) to \(f(x) + k\) and \(f(x + h)\text{.}\)

Write the formula for a basic function which has been changed by these transformations.

Sketch the graph of a basic function which has been changed by these transformations.

Subsection5.1.1Prep Activity

In preparation for this activity, we should revisit function evaluation with an eye toward describing changes to the input or the output.

Each of the following problems involves something you have already learned. These are skills we will need in the rest of this activity.

In this section, we will see the effects of changing the input or the output of a function, particularly as relates to the graph. We will follow a single story that will lead us through most of the main points of this section.

Example5.1.5

In preparation for the swimming competition, a swimmer jumped off a diving board into a swimming pool below. Below is a graph of her height above the water as a function of time.

You can see some basic information from the graph:

For the swimmer’s first jump, the function \(h = f(t)\) models her height above the water as a function of time. Use function notation to describe this change in height. See the next exercise.

On the day of the swimming competition, our swimmer was standing on the regular (\(3\) foot) diving board, waiting for the starting whistle. However, she was thinking so hard about her math class (which she loves), that she missed the sound of the whistle, and ended up jumping off the board \(2\) seconds late! See the next exercise.

It may have surprised you to see that shifting the graph to the right corresponded to subtracting a number from the input. To make sense of this, consider one of the other swimmers in the competition who jumped when the whistle sounded. Their height function would be given by \(h = f(t)\text{.}\)

If we wanted to find our swimmer's height at, say, \(3\) seconds after the whistle sounded, it would be the same height as this other swimmer had at only \(1\) second after the whistle — everything for our swimmer happened \(2\) seconds late. In other words, to find our swimmer's height after \(t\) seconds, this is the same as the other swimmer had \(2\) seconds before. So for our swimmer, her height is given by:

\begin{equation*}
h = f(t - 2)
\end{equation*}

The next two exercises will complete our story about the swimmer. In them, you will write formulas for her height given different diving situations.

Now we will see the effect of making a change to the output of a function by manipulating its formula. You will be able to describe what happens to the graph of a function after it has been changed from:

\begin{gather*}
y = f(x)\\
\text{to}\\
y = f(x) + k
\end{gather*}

Next, you will see what happens when we change the input of a function. You will be able to describe what happens to the graph as you change the formula from:

\begin{gather*}
y = f(x)\\
\text{to}\\
y = f(x+h)
\end{gather*}