lesson 12.3.1 version...

© 2013 THE CARNEGIE FOUNDATION FOR THE ADVANCEMENT OF TEACHING A PATHWAY THROUGH STATISTICS, VERSION 2.5, STATWAY® -‐ STUDENT HANDOUT

STATWAY® STUDENT HANDOUT

Lesson 12.3.1 Multiple Representations of Exponential Models

INTRODUCTION In Module 12 we have seen how linear functions and linear inequalities can be used to solve real world problems. Linear functions have the key property of a constant rate of change. This means that every increase in x of 1 unit causes the exact same change in y. In this lesson we will reexamine exponential functions and identify their key properties. Comparing Investments Imagine that you want to invest some money. You are considering two investments.

§ Investment 1: Bascom Blackwell offers an investment that doubles your money every four years.

§ Investment 2: Cassandra Cooper offers an investment that increases 18.92% each year.

The amount of money you plan to invest is between $100 and $1000. This initial amount of money will remain in either Investment 1 or Investment 2 for 8 years. Decide on an initial amount that you will invest and write the initial amount on the line below.

Amount to invest at the beginning: 1 To compare the two investments write the initial amount you chose for year 0 in the table below.

Complete the table using the initial amount you selected as your starting point. Calculate the value of each investment over the 8 year period. You do not need to find a value for the shaded part of the table, instead find the value of the Investment 1 at years 0, 4, and 8, and find the value of Investment 2 every year from 0 to 8. Be sure to round your answers to three decimal places Hint: Even though your initial amount may be different from your classmates, the steps to find the values of the investments will be the same.

STUDENT NAME DATE

STATWAY® STUDENT HANDOUT | 2



Years since beginning of investment

Investment 1 Investment 2

0 1 2 3 4 5 6 7 8

2 What do you notice about the value of each investment at the end of the 8 years? 3 Let’s focus on Investment 2. Can Investment 2 be modeled with a linear function? Does it increase

by the same amount each year? 4 Develop a formula for the value of Investment 2 over time. Let B represent the value of

Investment 2 and let t represent the number of years since the investment began. Hint: the formula must include the initial amount of the investment and the yearly growth rate.




NEXT STEPS In Module 4 we learned about exponential curves. An exponential curve has the general form:

! = ! ∙ !!

The numbers A and b in this exponential function have the following key properties:

• A is the y-‐intercept of the curve. It is the y-‐value when x = 0. It is also the initial value.

• If b > 1, b is the growth factor. If 0 < b < 1, b is the decay factor.

We can use this general form to determine an exponential function for real-‐world situations based on written descriptions. Situations that use exponential functions often involve percent increases or decreases. To translate a written description to an exponential function, we must identify the initial value and the growth factor or decay factor. Creating an Exponential Function using a Growth Factor and an Initial Amount In the previous example, Investment 2 grew by 18.92% each year. This is an example of exponential growth. The percent increase each year is 18.92%. This means that each year, the value of the investment increases by 18.92% of the value at the beginning of that year. The growth factor is equal to 100% + !"#$"%& !"#$%&'% = 100% + 18.92% = 118.92% = 1.1892. Each year the value of the investment is multiplied by 1.1892. Recall that we convert the percent to a decimal for arithmetic purposes. To do this we move the decimal left two places. For example, suppose the initial amount of your investment is $300. If the yearly growth factor is 1.1892, the investment will increase by 18.92% in the first year. At the end of the first year (1 year since the beginning) the investment will be worth 300 1.1892 = $356.76. This is shown in the second row of the table on the next page. In the second year, the investment will again increase by 18.92%. At the end of the second year (2 years since the beginning) the investment will be worth 356.76 1.1892 = $424.26. Rather than using the value after 1 year (356.76), we will instead use the initial value (300) to find the value of the investment at the end of the second year. To do this, we multiply the initial amount (300) by the growth factor twice. 300 1.1892 1.1892 This and more is shown in the third row of the table below. We can use this pattern to determine the value of the investment at any year. To determine the value of the investment after t years, we must multiply the initial amount by the growth factor t times. This generalization leads to the exponential function, ! = 300(1.1892)!, where A is the amount of the investment t years since the beginning. This is shown in the last row of the table.




Years since beginning

Value of Investment

Value of Investment in Expanded Form

Value of Investment

0 300 300 300 1 300 1.1892 300(1.1892)! 356.76 2 356.76 1.182 300 1.1892 1.1892 = 300(1.1892)! 424.26

3 424.26 1.182 300 1.1892 1.1892 (1.892) = 300 1.1892 ! 504.53

t

300 (1.1892)(1.1892)… (1.892) 300 1.1892 !

Decay Factor The previous problem was an example of exponential growth. In this next example we will work on creating an exponential function with a decay factor. We will work on calculating percent decrease over time. Whenever you take medication you are instructed to take a specific amount of medication every number of hours. The reason for this is that the amount of medicine in our bloodstream decreases exponentially over time. Suppose the amount of drug in a person’s bloodstream decreases by 30% each hour. Another way to think about this is that you have a certain amount of the drug in your system at the beginning of the hour. By the end of the hour, there will be 30% less of that drug in your body. In this example, the decay factor is equal to 100% − !"#$"%& !"#$"%&" = 100% − 30% = 70% = 0.70. Each hour, 30% of the drug is lost, and 70% of drug remains. If the initial amount of drug is 6 mg, and the hourly decay factor is 0.70, the function for the exponential model is ! = 6(0.70)! where h represents the number of hours since the drug was administered. TRY THESE Use your knowledge of growth and decay factors to write exponential functions for the following situations. 5 Cobalt 60 is a radioactive element which decays at 14% per year. If a lab has 35 grams of Cobalt 60

on Jan 1, 2003, write a function for the amount, A, of Cobalt 60 they have t years after Jan. 1, 2003.




6 Statisticians have estimated that the world population increases by about 1.2% each year. In September of 2012, the world population was estimated to be approximately 7.04 billion people. Write a function for the estimated world population, P, t years after September 2012.

7 Lake Merritt has been chemically polluted by the farms in its watershed and is now in the process

of being cleaned. When the clean-‐up of Lake Merritt started on June 1, 2011, the concentration of chemical pollutant was 810 microliters per milliliter of water. The mathematical model for this cleanup says that, for each month after they started, the amount of the pollutant in the lake is three-‐quarters of the amount of the previous month. Write the function for the concentration of pollutant in the water, P, at t months after June 1, 2011.

8 The function, ! = 150,000 1.035 !, describes the population of a town, P, at t years after 2000.

Write a description of the exponential growth or exponential decay process in the context of the size of the town’s population. Think about the starting population of the town and how fast the population is increasing or decreasing.

9 The function, ! = 26000 0.85 !, describes the value, V, of a new vehicle t years after it is

purchased from a car dealership. Write a description of the exponential growth or exponential decay process in the context of the vehicle’s value.

10 Describe in words a specific example of something that grows exponentially. Think about

something that increases by the same percent every hour or month or year.




11 Exchange your description with a partner and then find the function for your partner’s written description.

12 Describe in words a specific example of something that decays exponentially. Think about

something that decreases by the same percent every hour or month or year. 13 Exchange your description with a different partner and then find the function for your partner’s

written description. NEXT STEPS Doubling Time At the beginning of this lesson we saw two different descriptions for the same exponential process. The first description, “doubling every four years”, provides information on the amount of time required for the investment to double. This is an example of a doubling time. The second description, “increases by 18.92% each year”, provides information on the yearly growth rate. We will now find an exponential function for Investment 1 when the only information we are given is the doubling time. Suppose Investment 1 started with an initial amount of $1000. Let’s examine how Investment 1 changes over each 4-‐year period. Let n be the number of 4-‐year periods after the beginning of the investment. We will double the amount during each of those periods. Here’s a table to illustrate the growth of the investment over several 4-‐year periods.




t = years since beginning n = number of four-‐year periods since beginning

A = amount in Investment 1

0 0 1000 4 1 1000·∙2 = 1000·∙21 = 2000

8 2 1000·∙2·∙2 = 1000·∙22 = 4000

12 3 1000·∙2·∙2·∙2 = 1000·∙23 = 8000

4n n 1000·∙2n

The last row of the table shows that the amount of money, A, in Investment 1 after n 4-‐year periods since the beginning of the investment is:

! = 1000(2)! . TRY THESE 14 Use the pattern in the n and t columns of the table above to write an equation that shows the

relationship between n and t. 15 Use the equation relating n and t in question 14 to replace n with t in the function for A. This will

give you a function for the amount of Investment 1, A, t years since the beginning of the investment.

16 Fill in the table below by finding the value of the Investment 1 at each year specified in the table.

Round your answers to one decimal place.

t = years since beginning A = amount in Investment 1 0 1 2 4 6 8




17 Graph the amount of the investment on the coordinate plane below. Does the shape of the graph show an exponential growth relationship?

18 Use the graph to estimate how long it will take for the investment to reach $3000. It will help if

you connect the points with a smooth curve. When we are given the doubling time of an exponential process, and the initial amount, we can create an exponential function to model what is happening. In the following examples, we will use an exponential function to determine the percent increase that occurs during each unit of time. 19 A biology research scientist is growing bacteria in a petri dish in a lab. His model says that under

these conditions, the bacteria will double every 3 days.

A If the culture starts with 400 bacteria, write a function for the number of bacteria, N, after t days. Use your answer to 15 to help you with this question.

80 1 2 3 4 5 6 7

4,000

0

400

800

1200

1600

2000

2400

2800

3200

3600


Am

ount

of

inve

stm

ent

($)




B Use your function to find the number of bacteria after 6 days. Is your function correct? Think about how many times the bacteria should double in 6 days.

C We know the amount of bacteria doubles every three days, but how much does it increase

each day? To answer, this question, we can calculate the percent increase. The percent increase during the first day can be calculated by the following formula:

% !"#$%&'% = !"#$%&'" !"#$% !" !"# 1 − (!"#$%&'" !"#$% !" !"# 0)

(!"#$%&'" !"#$% !" !"# 0)

D Calculate the percent increase from day 0 to day 1. E Calculate the percent increase from day 1 to day 2.

F Calculate the percent increase from day 4 to day 5.

G What do you notice about the percent increase?

F Use the percent increase to write an exponential function for the amount of bacteria, A, in the petri dish after t days. Write the function using the growth factor, instead of the doubling time.




TRY THESE Half-‐Life 20 Cesium 137 is a radioactive chemical element that is released into the environment after nuclear tests

and nuclear disasters. It has a half-‐life of 30 years. This means that every 30 years the amount of Cesium 137 will decrease by half. To find out how much remains we multiply by ½.The McIntosh lab had 8 grams of Cesium 137 on January 1, 1975. The amount of Cesium 137, A, remaining can be described by the exponential function

! = 812

!!" ,

Where t is the number of years since January 1, 1975.

A How much did the McIntosh lab have on Jan 1, 1992?

B When would this Cesium 137 have been reduced to 6 grams? Complete the table and the graph below to help you answer this question.

t A 0

2

4

6

8

10

12

14

16

18

20




C As you can see from the graph, Cesium 137 decreased each year. What is the percent decrease each year?

D Write the exponential function for the amount of Cesium 137 remaining after t years using the decay factor. Hint: Calculate the decay factor using the percent decrease.

29 What information on an exponential process do you need to create a function for an exponential

model?

200 2 4 6 8 10 12 14 16 18

8

2

3

4

5

6

7

Years since January 1, 1975

Am

ount

of

Ces

ium

13

7




TAKE IT HOME 1 A small town had a population of 12,305 in 1995. The population of the town grew by 5% each

year over the next decade.

A Find a function for the population of the town, P, for t = years since 1995.

B Find the population of the town in 2001.

2 The function ! = 550 0.90 !, measured in grams, describes the amount of a radioactive

substance remaining t years after a study began. Write a description of this relationship in words.

3 The function ! = 2500 1.02 !, measured in dollars, describes the amount of money in the bank t years after an account was opened.

A Write a description of this relationship in words.

B Describe what you would do to answer the question “When will the account have $4000 in it?” (You do not have to answer that question – just describe your plan to answer it.) In your answer, think about how you would organize and display the information. Also think about how you would label the information.

STUDENT NAME DATE




4 Suppose the amount of money in a bank account increases by 6% per year.

A Write a function for A, the amount of money in the account t years after $1500 was deposited.

B How much money is in the bank after 7 years?

C Graph A, the amount of money in the account during the first 5 years. Is this an exponential relationship?

5 The function ! = 13523 2 !

!" describes the population of a town at t years after 1990.

A Write a description of this relationship in words. In your description, think about how many people live in the town and the rate of growth.

50 1 2 3 4

2,000

1,400

1500

1600

1700

1800

1900


Am

ount

of

mon

ey (

$)




B When was the population of the town about 15,000? Fill in the table below to help you answer this question.

T P 0

1

2

3

4

5

6

7

8

C The population of the town is growing each year? What is the percent increase each year?

D Create an exponential function based on the growth rate for the population of this town t

years after 1990.

6 The radioactive isotope Sulfur-‐35 has a half-‐life of 87 days. Suppose we have 15 grams of this chemical element. A function for the amount of Sulfur-‐35, A, remaining after t days is

! = 200 !!

!!".

A Find the amount remaining after 50 days and after 100 days.




B Find the percent decrease of Sulfur-‐35 each day.

C Create an exponential function based on the decay rate for the amount of Sulfur-‐35 remaining after t days.

+++++

This lesson is part of STATWAY®, A Pathway Through College Statistics, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the Statway® Networked Improvement Community, please visit carnegiefoundation.org.

+++++

Statway® is a trademark of the Carnegie Foundation for the Advancement of Teaching. It may be retained on any identical copies of this Work to indicate its origin. If you make any changes in the Work, as permitted under the license [CC BY NC], you must remove the service mark, while retaining the acknowledgment of origin and authorship. Any use of Carnegie’s trademarks or service marks other than on identical copies of this Work requires the prior written consent of the Carnegie Foundation.

This work is licensed under a Creative Commons Attribution-‐NonCommercial 3.0 Unported License. (CC BY-‐NC)

lesson 12.3.1 version...

Documents