Precalculus: An Active Reading Approach: Elementary Functions

Subsection 5.1.1 Prep 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.

For any function \(y = f(x)\text{,}\) remember that \(x\)-values are inputs, and \(y\)-values are outputs. If we change this equation to

then we are really just adding the number \(10\) to all of the outputs for the original function \(f\text{.}\)

In the next exercise, you will describe changes to either the input or the output of a function.

Lastly, you will write equations which match verbal descriptions of changes to the input and/or output.

Subsection 5.1.2 The Swimmer

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.

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:

1. How high was the diving board?

2. When did the swimmer reach the water?

3. What was her maximum height above the water?

Solution

1. The diving board was \(3\) feet high, as this is the swimmer's height above the water at \(t = 0\) seconds.

2. The swimmer reached the water at approximately \(1.1\) seconds, according to the graph.

3. Her maximum height above the water was approximately \(6.5\) feet, according to the graph.

After that dive, the swimmer decided to try out the high-dive board, which is \(7\) feet higher than the first board. See the next exercise.

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.

Now more information about that first dive: The formula for the swimmer's height above the water after diving off the regular board is given by:

(You are not expected to know how this formula was made.)

Confirm that this formula really works by graphing it on your calculator.

In the next exercise, you will change this formula so it represents the swimmer's height when she used the high-dive board.

Nice work so far, but our swimmer isn't done yet — she has two other boards to try out. See the next exercise.

In the next exercise, summarize what you now know about the effect of adding to, or subtracting from, the output of a function.

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:

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

Subsection 5.1.3 Seeing the shifts

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:

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:

Finally, we will see an application of using vertical and horizontal shifts to quickly perform a familiar task: Finding the equation of a line