This Warm-up prompts students to think about how changing the graphing window influences what you can see and understand about an exponential function.

Invite students to share some suggestions for a graphing window that are more helpful and those that are less helpful. Ask them to explain their reasoning.

Help students understand that, as a general rule, the values to show for a graph are usually determined by the quantity we are interested in studying. The values need to be selected carefully so that:

• Data of interest shows up on the graph.
• Interesting trends (of increase or decrease) are as visible as possible.

If time permits, discuss:

• “Why does the graph show such a sharp decrease between values of 0 and 2 and then start to flatten out?” (The decay factor is 0.2, which means that whenever increases by 1, it loses 0.8 of its quantity and keeps only 0.2. That decay is more apparent when the quantity is larger. As it gets smaller, and given the scale of the graph, it is harder to see the change. For example, when increases from 0 to 1, decreases by 320, because . But when increases from 2 to 3, decreases by  (or 10.8), which is a much smaller drop.)

In this activity, students examine the successive heights that a tennis ball reaches after several bounces on a hard surface, and they consider how to model the relationship between the number of bounces and the height of the rebound. To do so, they need to determine the growth factor of successive bounce heights. Because some data is provided here, students engage in only some aspects of mathematical modeling. To engage students in the full modeling cycle that includes data gathering, consider asking students to measure the bounce heights of a ball, as suggested in the next optional activity.

Real-world data is often messy, and that is the case for the data provided here. While each successive bounce height is about half of the preceding height, there is variation in the data, with the largest factor being a little more than 0.55 and the smallest a little less than 0.47.

Monitor for students who:

• Decide that an exponential model is not appropriate because the growth factor is different from bounce to bounce.
• Make a rough approximation for the growth factor, for example, observe that each bounce height, , is about half the preceding bounce height: .
• Find and use the growth factor from the first two points: .
• Take an average of the successive quotients: .
• Use graphing technology to generate a regression equation: .

Have students present in this order to support more precise methods for finding a model function to fit data.

Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).

This activity, designed for an extra class period, gives students an opportunity to gather and analyze data for bounce heights for multiple balls. Each ball should be sufficiently bouncy to allow measurement of at least 4 bounces. Good examples include tennis balls, basketballs, super balls, golf balls, and soccer balls. It will also be important to find a surface that is hard, flat, and level. Any padding will dampen the bounces, and any slant or irregularity on the surface will affect the direction of the bounce. A tiled or concrete floor, or a flat and paved surface outdoors should work. Students will need measuring tapes and may need some practice gathering the data.

Notice how students record the bounce heights. Recording these heights to the nearest inch or centimeter will already be challenging, and anything beyond that is too much precision. This activity is a good opportunity to choose a degree of precision appropriate to the context and to the measuring device used (MP6).

As in the previous task, monitor for how students process their data and decide on an appropriate factor to quantify the bounciness of each ball. Students should now be comfortable with the fact that the data is not exactly exponential but may still choose different ways for deciding on an appropriate exponential decay factor.

Here is a typical rebound factor for several types of balls:

• Tennis ball: 0.5 to 0.6
• Super ball: 0.9
• Basketball: about 0.6

Making graphing or spreadsheet technology available gives students an opportunity to choose appropriate tools strategically (MP5).

This activity continues to examine exponential decay in the context of successive ball bounces. Students use the given data to calculate a rebound factor and use it to write a function that models the relationship between number of bounces and bounce heights. They then use the function to answer questions about the ball and its bounces.

Students also think about the domain of the function and address the fact that this is a discrete context. It does not make sense to examine of a bounce, so the graphs should not be continuous.