Exponential Relationships

Pool Closing

Most times rumors are inaccurate comments spread about someone or something by word of mouth. Suppose that to study the spread of information through rumors, a student started the following rumor at 4:00 pm, one afternoon by calling a friend:

Due to an unexpected water leak, the pool will be closed for the rest of the week.

The next day, students were surveyed at school to find out how many heard the rumor and when they had heard it. How fast do you think this rumor would spread?

The graphs below show three possible patterns with the rate at which the pool- closing rumor spread.

Time (Hours)

  1. How would your describe the rate of rumor-spread for each graph?

Series 1: ______

______

Series 2: ______

______

Series 3: ______

______

  1. Which pattern of spread is most likely if the students start the rumor on the television or radio? Why?
  1. Which pattern of spread is most likely if the students start the rumor on the telephone or by word of mouth? Why?

Pool Party Raffle

After the rumor was stopped, the pool decided to have a raffle to welcome people back. Jennifer is making tickets for the pool party raffle. She starts by cutting a sheet of paper in half. She then stacks the two pieces and cuts them in half. She repeats this process, creating smaller and smaller pieces of paper to use as raffle tickets.

Jennifer wants to find a way to predict the number of tickets after any number of cuts.

After each cut, Jennifer counts the tickets and records the results in the table below.

Cuts / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
Tickets / 1 / 2
  1. Cut a sheet of paper as Jennifer did, and count the raffle tickets after each cut. Complete the table above to show the number of tickets after 2 cuts, and so on.
  1. When Jennifer looked at the table, she saw a pattern in the way the number of tickets changed with each cut? Find and use this pattern to extend your table to show the number of tickets for up to 10 cuts.
  1. Graph the (number of cuts, number of tickets) data for 1 through 10 cuts.

  1. As the number of cuts increase, how does the number of tickets change?
  1. What does this pattern of change tell you about the number of tickets produced for each cut?

The pattern you have just investigated is called an exponential growth pattern. Exponential growth patterns of change can be modeled using rules involving exponents.

When you found the number of tickets made by 10 cuts, you probably found yourself multiplying long strings of 2s. Instead of writing out long product strings of the same factor, you can use exponential form, so rather than writing , we can write , where 2 is the base and 7 is the exponent. We say that 128 is the standard form for writing .

  1. What is the shorthand way of writing the calculations you found for the values in your table? Does it work in finding all the values in the table? Does it hold true for all the values in your graph?
  1. If Jennifer made 20 cuts, how many tickets would she have? ______
  1. How many tickets would she have if she made 30 cuts? ______
  1. Write a rule using exponents that could be used to calculate the number of tickets for any cut, without knowing the amount made from the previous cut.

You can use your graphing calculator and the exponential rule for the number of tickets made based on the number of cuts, x, to make tables and graphs of the pattern formed by making raffle tickets. Enter the rule in the “Y=” list of your calculator, using the ^ key before the exponent. Graph the equation and make a table showing the number of tickets made for cuts 0 through 10. Sketch a copy of your graph and table below.

X / Y
0
1
2
3
4
5
6
7
8
9
10

  1. Find the number of tickets made from 15, 25, and 35 cuts. How did you find these values?

15 cuts: ______

  1. How many cuts are necessary to make 4096 tickets? Explain how you found this value.

More Exponential Growth

In studying exponential growth, it is common to refer to the starting point of the pattern as Stage 0 or the initial value.

  1. Use your calculator and the up arrow key to find each of the following values:

a. ______

b. ______

c. ______

d. ______

2. What seems to be the calculation for ? ______

  1. Use your calculator to make tables of (x, y) values for each of the following equations. Use values for x from 0 to 10.
x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
y
x / 0 / 1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9 / 10
y
  1. How are these tables different from the table with the ticket cuts?
  1. What patterns do you see in your tables that show how to model exponential growth from any starting point?
  1. If you see an equation of the form relating the variables x and y, what will the values of (a) and (b) tell you about the relation?
  1. The following tables show variables changing in a pattern of exponential growth. What equations will give rules for the patterns in the tables?

x / 0 / 1 / 2 / 3 / 4 / 5 / 6
y / 1 / 2 / 4 / 8 / 16 / 32 / 64

y= ______

x / 0 / 1 / 2 / 3 / 4 / 5 / 6
y / 3 / 6 / 12 / 24 / 48 / 96 / 192

y= ______

  1. Penicillin was discovered by observation of mold growing on biology laboratory dishes. Suppose a mold begins growing on a lab-dish, and when it was first observed, the mold covered only 1/8 of the dish surface. It appears to double in size every day. When will the mold cover the entire dish?
  1. Write a rule that models the penicillin scenario, and describe the shape of the graph it would make.

Rule: ______

Graph Description: