The graphs below show three possible patterns with the rate at which the pool- closing rumor spread. Spreading the Rumor. Series 1: Series 2:

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1 Number of People Exponential Relationships Pool Closing Most times rumors are inaccurate comments spread about someone or something b word of mouth. Suppose that to stud the spread of information through rumors, a student started the following rumor at 4:00 pm, one afternoon b calling a friend: Due to an unexpected water leak, the pool will be closed for the rest of the week. The next da, students were surveed at school to find out how man heard the rumor and when the had heard it. How fast do ou think this rumor would spread? The graphs below show three possible patterns with the rate at which the pool- closing rumor spread. Spreading the Rumor Time (Hours) 1. How would our describe the rate of rumor-spread for each graph? Series 1: Series 2: Series 3:

2 2. Which pattern of spread is most likel if the students start the rumor on the television or radio? Wh? 3. Which pattern of spread is most likel if the students start the rumor on the telephone or b word of mouth? Wh? Pool Part Raffle After the rumor was stopped, the pool decided to have a raffle to welcome people back. Jennifer is making tickets for the pool part raffle. She starts b 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 wa to predict the number of tickets after an number of cuts. After each cut, Jennifer counts the tickets and records the results in the table below. Cuts Tickets 1 2 a. 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. b. When Jennifer looked at the table, she saw a pattern in the wa the number of tickets changed with each cut? Find and use this pattern to extend our table to show the number of tickets for up to 10 cuts. c. Graph the (number of cuts, number of tickets) data for 1 through 10 cuts.

3 4. As the number of cuts increase, how does the number of tickets change? 5. What does this pattern of change tell ou about the number of tickets produced for each cut? The pattern ou have just investigated is called an exponential pattern. Exponential patterns of change can be modeled using rules involving exponents. When ou found the number of tickets made b 10 cuts, ou probabl found ourself multipling long strings of 2s. Instead of writing out long product strings of the same factor, ou can use exponential form, so rather than writing 2*2*2*2*2*2*2, we can write 2 7, where 2 is the base and 7 is the exponent. We sa that 128 is the standard form for writing What is the shorthand wa of writing the calculations ou found for the values in our table? Does it work in finding all the values in the table? Does it hold true for all the values in our graph? 7. If Jennifer made 20 cuts, how man tickets would she have? 8. How man tickets would she have if she made 30 cuts? 9. Write a rule using exponents that could be used to calculate the number of tickets for an cut, without knowing the amount made from the previous cut. You can use our 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 b making raffle tickets. Enter the rule in the Y= list of our calculator, using the ^ ke before the exponent. Graph the equation and make a table showing the number of tickets made for cuts 0 through 10. Sketch a cop of our graph and table below. X Y

4 a. Find the number of tickets made from 15, 25, and 35 cuts. How did ou find these values? 15 cuts: b. How man cuts are necessar to make 4096 tickets? Explain how ou found this value. More Exponential Growth In studing exponential, it is common to refer to the starting point of the pattern as Stage 0 or the initial value. 1. Use our calculator and the up arrow ke to find each of the following values: a. 3 0 b. 6 0 c d What seems to be the calculation for b 0? a. Use our calculator to make tables of (x, ) values for each of the following equations. Use values for x from 0 to 10. 3(2 x ) x (3 x ) x b. How are these tables different from the table with the ticket cuts? c. What patterns do ou see in our tables that show how to model exponential from an starting point? d. If ou see an equation of the form a(b x ) relating the variables x and, what will the values of (a) and (b) tell ou about the relation?

5 e. The following tables show variables changing in a pattern of exponential. What equations will give rules for the patterns in the tables? x = x = f. Penicillin was discovered b observation of mold growing on biolog laborator dishes. Suppose a mold begins growing on a lab-dish, and when it was first observed, the mold covered onl 1/8 of the dish surface. It appears to double in size ever da. When will the mold cover the entire dish? Initial After 1 da s After 2 das After 3 das After 4 das g. Write a rule that models the penicillin scenario, and describe the shape of the graph it would make. Rule: Graph Description: