1

Comparing Linear and Exponential Functions (Day 6–1)

Recall: Fill in the table below and then graph.

a) What type of function is it?

b) What is the average rate of change?

c) What is the equation of the function?

1. Fill in the table and graph.

a) Is the function linear? Why or why not?

b) What is the pattern for )(xf ?

c) What type of function is it?

Linear Function: A function that has a ____________________________________. When looking

at a table, there should be a pattern of ______________________________ called a common

________________________.

Exponential Function: A function that has “x” as an ______________________. When looking

at a table, there should be a pattern of ______________________________ called a common

________________________.

x f(x)

1 –5

2 –2

3 1

4 4

5

6

7

x f(x)

1 2

2 4

3 8

4 16

5

6

7

2

2. Use the function 32)( xxg to fill in the table below and graph.

a) What type of function is this and why?

b) What is the domain?

c) What is the range?

3. Use the function

x

xg

2

1)( to fill in the table below and graph.

a) What type of function is this and why?

b) What is the domain?

c) What is the range?

4. Describe each of the following functions as either linear or exponential:

a) 23)( xxf b) 152 xy c) 7y

d) 53

2)( xxj e)

x

y

4

3 f) xxC )052.1(332,16)(

x g(x)

–3

–2

–1

0

1

2

3

x g(x)

–3

–2

–1

0

1

2

3

3

Classwork 6–1

1. Fill in the missing parts of each of the following tables and determine what type of

function they represent. Give a brief explanation to why it is that function.

a) b)

c) d)

2. Graph the function:

72)( xxf

x f(x)

–3 –10

–2 –5

–1 0

0

1

2

3 20

x f(x)

1 4

2 16

3 64

4

5 1024

6

7

x f(x)

2 512

3 256

4 128

5

6 32

7 16

8

x f(x)

–1

0 12

1

2 24

3

4 36

5

4

3. Match the letter of the appropriate equation and number of the appropriate context to

the following tables:

Equation ______ Equation _______ Equation _______ Equation _______

Context _______ Context ________ Context ________ Context ________

Equations:

A. 4012)( xxf

B. xxg 3)(

C. xxh )5(.64)(

D. xxp 5)(

4. Tina wants to make a rectangular sandbox that has a perimeter of 76 inches. The width is

five more than twice the length. Determine the dimensions of the sandbox.

x f(x)

0 0

1 5

2 10

3 15

4 20

5 25

6 30

x f(x)

0 64

1 32

2 16

3 8

4 4

5 2

6 1

x f(x)

0 40

1 52

2 64

3 76

4 88

5 100

6 112

x f(x)

0 1

1 3

2 9

3 27

4 81

5 243

6 729

Contexts:

1. The population of bacteria in a dish triples every day.

2. A waiter makes an hourly wage plus tips.

3. A speed walker walks at a constant speed.

4. Half of the teams are eliminated from a basketball

tournament every round.

5

Linear vs. Exponential Functions (Day 6–2)

1. Two equipment rental companies have penalty policies for returning their equipment

late:

Company 1: On day 1, the penalty is $5. On day 2, the penalty is $10. On day 3, the

penalty is $15 and so on, increasing by $5 each day the equipment is late.

Company 2: On day 1, the penalty is $0.01. On day 2, the penalty is $0.02. On day 3, the

penalty is $0.04 and so on, doubling in amount each additional day late.

Jim rented a digger from Company 2 because he thought it had the better late return

policy. The job he was doing with the digger took longer than he expected, but it did not

concern him because the late penalty seemed so reasonable. When he returned the

digger 15 days late, he was shocked by the penalty fee.

a) What did Jim pay to Company 2?

b) What would Jim have paid to

Company 1?

c) What type of function does the fees

for Company 1 represent and why?

d) What type of function does the fees for Company 2 represent and why?

6

2. A typical thickness of toilet paper is 0.001 inches. Seems pretty thin, right? Let’s see what

happens when we start folding toilet paper.

a) How thick is the stack of toilet paper after 1 fold?

After 2 folds?

After 5 folds?

b) Write a formula for the function that models the thickness of the folded toilet paper

after “x” folds.

c) The top of a building is 2500 feet high. If we fold the toilet paper 25 times, will it reach

to the top of the building?

FORM OF EXPONENTIAL FUNCTIONS: xaby OR xabxf )(

“a” is the ________________ value, population, number, etc. It is the number that you

are starting with.

“b” is the growth/decay _____________ . If b is greater than 1, it is a ____________.

If b is less than 1, it is a _____________.

“x” is usually refers to _____________ , unless otherwise stated.

3. A tennis tournament has 128 competitors. Half of the competitors are eliminated each

round. Write a function to represent the number of competitors that will be left after “x”

rounds. Then determine how many players will be left after 5 rounds.

7

Classwork 6–2

1. Kyla is given two options for how she is to be paid at her job. Option A is to be paid $1000

at the end of each month. Option B is to be given $5 to start the year, and then get paid

$10 at the end of the first month, $20 the second month, $40 the third, doubling the

amount she receives each month. .

a) Write a function that can represent the amount that Kyla would be paid in any month

with Option A.

b) Write a function that can represent the amount the Kyla would be paid in any month

with Option B.

c) How much would Kyla get paid during the 8th month using option B?

d) Which option would you rather have? Justify your answer.

For questions #2 – 5:

a) State whether the function is a growth or decay and why.

b) State the initial value.

2. ttc )75(.100)( 3. nnp )80.1(40)(

4. xxt )02.1(000,10)( 5. nnf )1(.50)(

8

6. There are 500 rabbits in Lancaster on February 1st. If the amount of rabbits triples every

month, write a function that represents the number of rabbits in Lancaster after “m”

months.

How many rabbits are there in Lancaster on August 1st?

7. Graph the function 62

1)(

x

xf .

8. Graph the following inequality:

xy 2164

9

Exponential Growth (Day 6–3)

xaby

“a” is the ________________ value, population, number, etc.

“b” is the growth/decay _____________ .

“b” will either be 1 + rate (growth) or 1 – rate (decay).

Recall: To turn a percent to a rate move decimal place ___________________, or

divide by ________.

“x” is usually refers to _____________ , unless otherwise stated.

1. The population of the country of Oz was 600,000 in the year 2010. The population is

expected to grow by a factor of 5% annually. The annual food supply of Oz is currently

sufficient for 700,000 people and is increasing at a rate which will supply food for an

additional 10,000 people per year.

a) Write a formula to model the population of Oz. Is your formula linear or exponential?

b) Write a formula to model the food supply. Is the formula linear or exponential?

2. In the 2000-2001 school year, the average cost for one year at a four-year college was

$16,332, which was an increase of 5.2% from the previous year. If this trend were to

continue, write a function to model the cost, )(xC , of a college education x years from

2000 and:

a) Find )4(C . Explain what )4(C represents in words.

b) If this trend continues, how much would parents expect to pay for their new born

baby’s first year of college? (Assume the child would enter college in 18 years.)

10

3. In 1993, the population of New Zealand was 3,424,000, with an average annual growth

rate of 1.3%. Suppose that this growth rate were to continue.

a) Express the population P as a function of n, the number of years after 1993.

b) Estimate New Zealand’s population in the year 2010.

4. Do the examples below require a linear or exponential growth model? Write a function

that models the growth for each case.

a) A savings account that starts with $5000 and receives a deposit of $825 per month.

b) The value of a house that starts at $150,000 and increases by 1.5% per year.

c) An alligator population starts with 200 alligators and every year, and it grows by a

factor of 7

9 each year.

d) The temperature increases by 2°F every hour from 8:00 a.m. to 4:00 p.m. for a July day

that has a temperature of 66°F at 8:00 a.m.

11

Classwork 6–3

1. A three-bedroom house in Burbville was purchased for $190,000. If housing prices are

expected to increase by 1.8% annually in that town, write a function that models the

price of the house in t years. Find the price of the house in 5 years.

2. In 1995, there were 85 rabbits in Central Park. The population increased by 12% each

year. How many rabbits were in Central Park in 2005?

3. The value of an early American coin increases in value at the rate of 6.5% annually. If the

purchase price of the coin this year is $1,950, what is the value to the nearest dollar in 15

years?

4. In 1985, there were 285 cell phone subscribers in the small town of Centerville. The

number of subscribers increased by 75% per year after 1985. How many cell phone

subscribers were in Centerville in 1994?

12

5. Ben & Jerry are going to Dairy Queen to order ice cream. They only brought dimes,

quarters, and half dollar coins with them. In order to pay their bill of $3.85, they used 2

more half dollar coins than quarters and also used twice as many dimes than quarters.

How many of each coin did they use to pay?

6. In a piggy bank there are only dimes and nickels. If all together there are 40 coins that

total $2.50 in value, how many of each coin are there in the bank?

13

Exponential Decay (Day 6–4)

Recall: The cost of Bob’s house in 2005 was $220,000. If his house appreciates in value at a

rate of 3.5% every year, what will the price of his house be in 2015?

Example 1: A ball is dropped from a height of 100 centimeters. When it hits the ground and

rebounds, the new height won’t be as high as it was at first. The height after each bounce

decreases by a rate of 20% from the previous height. What would the height of the ball be

after the 4th bounce? Use the table below to help you find your answer.

1. What did you do to find the height of the ball after each bounce?

2. What is the ratio between the value after 1 bounce and the start height? What is the

ratio between the height after 2 bounces and the height after 1 bounce? What about

the height after 3 bounces and height 2 bounces? Etc…

3. What does the value 0.80 have to do with a 20% height loss?

Number of bounces, x Height after x

bounces 20% of height lost

New height after

the bounce

0 100 20 80

1 80

2

3

4

14

Recall: In the last question, the equation bbH )80(.100)( that models the height after each

bounce looks quite similar to the formula/equations we were using in the last two lessons

for exponential growth. Is the height of the ball’s bounce increasing?

What about the equation could tell you that it is not a growth model?

Practice: For each of the following:

a) State whether it a growth or decay.

b) State the initial value.

c) State the percent rate of growth or decay.

1. ttf )5.1(2)( 2. ttb )8(.5)(

3. ttg )45(.2

1)( 4.

t

th

3

22000)(

5. Ryan bought a new computer for $2,100. The value of the computer decreases by 50%

each year. After what year will the value drop below $300?

6. Kelli’s mom takes a 400 mg dose of aspirin. Each hour, the amount of aspirin in a person’s

system decreases by about 29%. To the nearest tenth of a milligram, how much aspirin is

left in her system after 6 hours?

15

Classwork 6–4

For questions #1-4:

a) State whether it a growth or decay.

b) State the initial value.

c) State the percent rate of growth or decay.

1. ttc )75(.100)( 2. nnp )80.1(40)(

3. xxt )02.1(000,10)( 4. nnf )1(.50)(

5. A huge ping-pong tournament is held in Beijing, with 65,536 participants at the start of the

tournament. Each round of the tournament eliminates half the participants.

a) If r represents the number of participants remaining after r rounds of play, write a

formula to model the number of participants remaining.

b) Use your model to determine how many participants remain after 10 rounds of play.

c) How many rounds of play will it take to determine the champion ping-pong

player?

16 0 2 4 6 8 10 12

# of years

Va

lue

of

eq

uip

me

nt

30,000

60,000

90,000

6. Kathy plans to purchase a car that depreciates at a rate of 9.2% per year. The initial cost

of the car is $21,000. Determine the value of the car after 3 years.

7. A construction company purchased some equipment costing $300,000. The value of the

equipment depreciates at a rate of 14% per year.

a) Write a formula )(tC that models the value of the equipment.

b) What is the value of the equipment (to the nearest dollar) after:

i) 2 years?

ii) 4 years?

iii) 6 years?

iv) 8 years?

v) 10 years?

c) Graph the points from part b on the grid below.

t C(t)

2

4

6

8

10

17

Exponential Application Word Problems (Day 6–5)

1. An oil spill is spreading such that its area is given by the exponential function ttA )15.1(250)( , where A is the area in square feet and t is the time that has elapsed in

days.

a) How large was the oil spill initially?

b) By what percent is the oil spill increasing each day?

2. A population of 50 fruit flies is increasing at a rate of 6% per day. Which of the following is

closest to the number of days it will take for the fruit fly population to double?

(1) 18 (3) 12

(2) 6 (4) 28

3. Water drains out of a pool, the depth of the water decreases at a constant rate of 20%

per hour. The depth of the water, when the draining begins, is 12 feet.

Write an equation that gives the depth, D, of the water in the pool as a function of the time

in hours it has been draining, t.

What is the depth of the pool in 5 hours?

It’s safe to cover the pool after it reaches a depth of 1 foot or less. What is the minimum

number of whole hours that we should wait to cover the pool?

Explain how you found your answer.

18

4. The value of an early American coin increases in value at the rate of 6.5% annually. If the

purchase price of the coin this year is $1,950, what is the value to the nearest dollar in 15

years?

5. A population of fruit flies is growing at a constant rate of 6% per hour. The population

starts started with 28 flies.

a) Write a formula that models the population, P, as a function of the time in hours, t.

b) What will be the population of flies in 2 days?

6. Which of the following equations would model an exponential quantity that begins at a

level of 16 and decreases at a constant rate of 8% per hour?

(1) tQ )92.0(16 (3) tQ )08.1(16

(2) tQ 92.016 (4) tQ )7(16

7. The amount, A, in grams of radioactive material that is decaying can be modeled by ddA )88.0(450)( , where d represents the number days since it started its decay.

a) What percent is the material decaying per day?

b) Given an interpretation of the fact that .75)14( A

19

Classwork 6–5

1. Tabitha purchased a house in 2002 for $179,000. The house is expected to appreciate in

value about 15% per year. Which equation could be used to find the house’s value, V,

after t years?

(1) tV )85(.000,179 (3) tV )15(.000,179

(2) tV )85.1(000,179 (4) tV )15.1(000,179

2. If a radioactive substance is quickly decaying at a rate of 13% per hour approximately

how much of a 200 pound sample remains after one day to the nearest tenth of a

pound?

3. A mouse population starts with 2,000 mice and grows at a rate of 5% per year. The

number of mice after t years can be modeled by the equation, ttP )05.1(2000)( . What is

the rate of change between the second year and the fifth year rounded to the nearest

whole number?

(1) 116 (3) 2205

(2) 348 (4) 2553

4. In 2006, the number of wolves in a wildlife preserve is 5,400. The average growth rate is

3%. If this growth rate continues, how many wolves will there be in 2016?

20

5. Tammy is saving money in a bank that charges a fee if the account is inactive for an

extended period of time. The equation xy )75(.1250 represents the value, y, of one

account that was left inactive for a period of x years. Which statement below is true?

(1) 1250 is the y-intercept, it is the amount of money in the account after x years

(2) 1250 is the y-intercept, it is the amount of money in the account initially

(3) 0.75 is the percent of money in the account initially

(4) 0.75 is the percent of money in the account after x years

6. Rhonda deposited $3000 in an account in the Merrick National Bank, earning 4.2%

interest, compounded annually. She made no deposits or withdrawals. Write an

equation that can be used to find B, her account balance after t years.

7. What is the average rate of change for the function xxxf 2)( 2 over the interval 2,3 ?

8. Which table represents a function? State the reason why.

(2) (4)

(1) (3)

21

Compound Interest (Day 6–6)

In history, banks calculated interest at the end of each year on the original amount either

borrowed or invested (the ______________). This type of interest was called ____________

_______________.

As time went on banks realized that there was another way to compute interest and make

even more money. They called this type of interest _____________ ____________.

To compute compound interest: Banks calculate the interest for the first period, ____________

it to the total, then calculate the interest on the _______ total for the next period, and so on.

Since we are calculating a rate % and it is growing to earn interest, the compound

interest formula models the exponential growth equation, xaby , just using different

letters to represent BANKING terms.

1. Sara had invested $800 in a savings account that paid 4.2% interest compounded

annually. How much money to the nearest cent was in the account after 4 years, if he

left the money untouched?

2. If Bailey invested money 4 years ago with an annual interest rate of 3.275% compounded

annually, and it is valued at $11,260, how much money did she initially invest?

Compound Interest: trptA )1()( A(t) = amount earned/owed after t years

P = principal amount borrowed or invested (starting a#)

r = interest rate % (must convert to a decimal)

t = time in years

22

3. If the savings in a bank account can be modeled by the function ttS )045.1(250)( . Which

of the following is true?

(1) The initial amount deposited was $250 and the interest earned was 45%

(2) The initial amount deposited was $2.50 and the interest earned was 4.5%

(3) The initial amount deposited was $250 and the interest earned was 4.5%

(4) The initial amount deposited was $2.50 and the interest earned was 45%

4. If $350 is placed in a savings account that earns 3.5% interest applied once a year, then

how much would the savings account be worth after 10 years?

5. Tammy has a balance of $5,620 in her savings account. She is moving this account to

another bank that has advertised an interest rate of 6.5% per year. Which of the following

equations would give Tammy’s account worth, W, as a function of the number of years, y,

it has been gaining interest.

(1) yW )65(.5620 (3) 5620065.1 yW

(2) yW )065.1(5620 (4) 562065.1 yW

23

Classwork 6–6

1. $325 is borrowed from a bank that charges 4% interest compounded annually. How

much is owed after 7 years; 20 years?

2. A youth group has a yard sale to raise money for a charity. The group earns $800 but

decided to put its money in the bank for a while. Calculate the amount of money the

group will have if:

a) Cool Bank pays compounded interest at a rate of 4%, but the youth group can only

leave the money in for 3 years.

b) Hot Bank pays compound interest at a rate of 3%, but the youth group can leave the

money in for 5 years.

c) Which bank is the better choice for their respective time restrictions, and how much

more money will they receive by choosing this bank?

3. If Piper invested money 7 years ago with an annual interest rate of 1.95% compounded

annually, and it is valued at $12,575, how much money, to the nearest cent, did she

initially invest?

24

4. Milton has his money invested in a stock portfolio. The value, )(xv , of his portfolio can be

modeled with the function xxv )78.0(000,30)( , where x is the number of years since he

made his investment. Which statement describes the rate of change of the value of his

portfolio?

(1) It decreases 78% per year

(2) It decreases 22% per year

(3) It increases 78% per year

(4) It increases 22% per year

5. The cost of airing a commercial on television is modeled by the function 900110)( nnC ,

where n is the number of times the commercial is aired. Based on this model, which

statement is true?

(1) The commercial costs $0 to produce and $110 per airing up to $900.

(2) The commercial costs $110 to produce and $900 each time it is aired.

(3) The commercial costs $900 to produce and $110 each time it is aired.

(4) The commercial costs $1010 to produce and can air an unlimited amount.

6. Solve for the solution to the following system graphically:

923

435

yx

yx

25

Introduction to Regressions (Day 6–7)

Regression Equation: an equation of a function (linear, exponential, etc..) that plotted points

follow or create. (calculator command creates equations)

Correlation Coefficient (r#): this number describes the strength of the equation in regards to

the plotted points. A PERFECT fit equation will have a r = 1 or r = –1 depending on _______ of

line or growth of an exponential curve.

(These concepts will be expanded further later in the year; these are just the basics needed for today)

Write an equation of the GRAPHED functions below.

Identify the type of function (Linear or Exponential)

Identify and write down the correct number of points needed for each function

Follow the steps above for the calculator.

1. 2.

HOW TO USE A CALCULATOR TO CREATE LINEAR AND EXPONENTIAL EQUATIONS

What do we need?

Linear Equations: 2 exact points from table, graph or word problem

Exponential Equations: 3 exact points from table, graph or word problems

Steps in Calculator: (need to turn Diagnostics On in calculator to get r#)

Put points in a list (STAT- EDIT)

Choose either LinReg (Linear Equ) or ExpReg (Exponential Equ) (STAT-CALC)

VERIFY it is a PERFECT fit equation by looking at r#. MUST = ±1 to be perfect

Write equation stated by filling in the appropriate values for the coefficients

EXPLAIN IN WORDS the steps you did to get the equation since no math was involved in writing the equations.

y

x

26

3. Write an equation of a line that goes through the points (–2, –11) and (3, 14).

4. Write the equation of the exponential function that goes through the following points:

(–1, 16) (2, 6.75) (1, 9)

5. Each day Toni records the height of a plant for her science lab. Her data are shown in

the table below.

The plant continues to grow at a constant daily rate. Write an equation to represent )(nh ,

the height of the plant on the nth day.

6. Write an exponential function )(xf for the table shown.

x 0 1 2 3 )(xf 13 39 117 351

27

7. The table below shows the amount of a decaying radioactive substance that remained

for selected years after 1990.

a) Write an exponential regression equation for this set of data, rounding all values to the

nearest hundredth.

8. Jackson is starting an exercise program. The first day he will spend 30 minutes on a

treadmill. He will increase his time on the treadmill by 2 minutes each day. Write an

equation for )(dT , the time, in minutes, on the treadmill on day d.

9. The illustration below shows of a pattern of blocks. Write an equation to represent this

model.

28

Classwork 6–7

1. Write an equation of a line that goes through the points (–7, 1) and (–1, –2).

2. Write an exponential equation that goes through (–4, 192), (–3, 48), and (–1, 3).

3. Write an equation of the graph below.

4. A sunflower is 3 inches tall at week 0 and grows 2 inches each week. Write an equation

of a function that could be used to determine the height, )(nf , of the sunflower in n

weeks.

5. An application developer released a new app to be downloaded. The table below

gives the number of downloads for the first four weeks after the launch of the app. Write

an exponential equation that models this data.

29

6. Caitlin has a movie rental card worth $175. After she rents the first movie, the card’s value

is $172.25. After she rents the second movie, its value is $169.50. After she rents the third

movie the card is worth $166.75. Assuming the pattern continues, write an eqation to

define )(nA , the amount of money on the rental card after n rentals.

7. The diagrams below represent the first three terms of a sequence.

Assuming the pattern continues, which formula determines )(nf , the number of shaded

squares in the nth term?

(1) 124)( nnf (3) 44)( nnf

(2) 84)( nnf (4) 24)( nnf

8. Write an equation of the function graphed below. Explain how you determined your

equation.