IM Algebra: Unit 5 Lesson 11

Unit 5 Lessons 11

Modeling Exponential Behavior

Fill in the Blanks

Let p be the function that gives the cost p(t), in dollars, of producing 1 watt of solar energy t years after 1977. This is a table showing the values of p from 1977 to 1987. Find the average rate of change between the years 1980 and 1984. Round your answer to the nearest whole number.

Fill in the Blanks

Here is a table that shows the number of coffee shops worldwide that a company had in its first 10 years, between 1987 and 1997. The growth in the number of stores was roughly exponential.

What is the average rate of change between 1990 and 1994?

Multiple Choice

When a graph shows exponential growth, is the average rate of change increasing or decreasing?

• Increasing

• Decreasing

• Neither, it remains constant

Multiple Choice

When a graph shows exponential decay, is the average rate of change increasing or decreasing?

• Increasing

• Decreasing

• Neither, it remains constant

Open Ended

Noah figured out that the average rate of change for this 5 year period was almost $300 per year. Noah thinks this describes the relationship in the data well. Do you agree or disagree with him? Explain your choice.

Average Rate of Change

When looking for the average rate of change, you can find the amount the data changed and divide it by the time span it changed over.

Linear functions will have the same rate of change (slope) no matter what interval you compare. Exponential functions will have different rates of change depending on what interval you are calculating.

Average rate of change is NOT the same as the growth/decay factor.

Lesson 5-11: Modeling Exponential Behavior

• Discuss the graphing window shown.

• Use a calculator to create the graph and make the information more meaningful.

• Compare with your neighbor

Open Ended

Here are measurements for the maximum height of a tennis ball after bouncing several times on a concrete surface.

1.) Which is more appropriate for modeling the maximum height h, in centimeters, of the tennis ball after n bounces: a linear function or an exponential function? Use data from the table to support your answer.

Open Ended

2.) Regulations say that a tennis ball, dropped on concrete, should rebound to a height between 53% and 58% of the height from which it is dropped. Does the tennis ball here meet the requirement? Explain your reasoning.

Open Ended

3.) Write an equation that models the bounce height h after n bounces for this tennis ball.

4.) About how many bounces will it take before the rebound height of the tennis ball is less than 1 centimeter? Explain your reasoning.

Open Ended

Here is a table of three different kinds of balls and their heights when bounced.

Which ball appears to be the bounciest? Which one appears to be the least bouncy? Explain your reasoning.

Open Ended

1.) For each ball, write an equation expressing the bounce height in terms of the bounce number n:

Ball 1: a =

Ball 2: a =

Ball 3: a =

2.) Explain how your equations could tell which ball is the bounciest.

3.) If the bounciest ball were dropped from a height of 300 cm, what equation would model its bounce height h?

Open Ended

The table shows some heights of a ball after a certain number of bounces.

1.) Is this more or less bouncy than the tennis ball in the earlier task? Explain your reasoning.

2.) From what height was the ball dropped? Explain your reasoning.

3.) Write an equation that represents the bounce height of the ball, h, in centimeters after n bounces.

Multiple Choice

Which graph would more appropriately represent the equation for h.

Multiple Choice

True or false. This ball will always have a higher bounce than the original tennis ball.

• True

• False

• Not enough info- we would have to test it to find out.

Summary- Modeling Exponential Behavior

• Sometimes data suggests an exponential relationship. For example this table shows the bounce heights of a certain ball.

• To find what fraction of the height remains after each bounce we can divide two consecutive numbers:

• 61/95  $\approx$  0.642, 39/61  $\approx$   0.639, and 26/39  $\approx$   0.667

• All of these quotients are close to 2/3. This suggests an exponential relationship and that the height is decreasing by a factor of 2/3. 

• The height of the ball could be modeled by the equation:  $h\ =\ 142\cdot\left(\frac{2}{3}\right)^n$  

Summary (cont'd)

• Here is a graph of the equation.

• This graph shows both the points from the data and the points generated by the equation, which can give us new insights.

• For example, the height from which the ball was dropped was not given but can be determined.

• For a second example, we can see that it will take 7 bounces before the rebound height is less than 10 centimeters.