ESSON #59 LINEAR FUNCTIONS AND MODELING ......Last lesson, we looked at situations that can be modeled by linear functions because the values increased or decreased by equal differences

1

COMMON CORE ALGEBRA II, UNIT #7 – LINEAR VS. EXPONENTIAL FUNCTIONS

eMATHINSTRUCTION, RED HOOK, NY 12571, © 2015

LESSON #59 - LINEAR FUNCTIONS AND MODELING

COMMON CORE ALGEBRA II

A linear function is any pattern where the function increases or decreases by the same numerical constant per

unit. It is a function where the rate of change is always constant.

The two most important qualities of a linear function are its starting point on the y-axis, known as the y-

intercept, and its constant rate of change or slope. These two qualities tell us where the function “starts” and

where it is “going.”

The most basic form, slope-intercept, is the one we will be using predominantly in this course. The point-slope

form is also included in the textbox below.

Exercise #1: Write a function to model each of the patterns in the tables below.

x 1 0 1 2 3

y 4 7 10 13 16

Exercise #2: Graph each equation. Equation Graph Equation Graph

2 7y x

2( ) 3

5f x x

x 0 1 2 3 4

y 180 160 140 120 100

TWO COMMON FORMS OF A LINE

Slope-Intercept: Point-Slope:

where m is the slope (or average rate of change) of the line and represents one point on the line.

2



Exercise #3: Write an equation for each graph. Equation Graph Equation Graph

Exercise #4: Dia was driving away from New York City at a constant speed of 58 miles per hour. He started

45 miles away.

Exercise #5: Two students have bank accounts.

Student A starts with $600 in her bank account and takes out $20 each month.

Student B starts with $900 in his bank account and takes out $50 each month.

(a) Create linear functions for amount of money, in

each account after x months.

Let A(x) =

Let B(x) =

Let x =

(b) Algebraically determine exactly how many

months it will take for Student A and Student B

to have the same amount in their accounts.

(a) Write a linear function that gives Dia’s

distance, D, from New York City as a function

of the number of hours, h, he has been driving.

Let h =

Let D =

(b) If Dia’s destination is 270 miles away from

New York City, algebraically determine to the

nearest tenth of an hour how long it will take

Dia to reach his destination.

3



Exercise #3: A factory is currently printing sci-fi paperback novels. Each day it costs $1000 to run the factory

and pay the workers. It also costs $3.50 per book to make the books.

Write a function, C(b), to model the total cost of producing b books each day.

The factory sells the books to a distributor for $4.75 per book.

Write a function, R(b), to model the revenue for the books each day.

(a) Use your graphing calculator to sketch and

label each of these linear functions for the

interval 0 1000b . Be sure to label both

axes with a scale.

Let b =

Let C(b) =

Let R(b) =

(b) Use your calculator’s INTERSECT command to

determine the number of sci-fi books, b, that

must be produced for the revenue to equal the

cost.

(c) If profit is defined as the revenue minus the

cost, create an equation in terms of w for the

profit, P.

4



LESSON #59 - LINEAR MODELING

COMMON CORE ALGEBRA II HOMEWORK

1. Write a function to model each of the patterns in the tables below.

x 1 0 1 2 3

y 4 7 10 13 16

2. Graph each equation. Equation Graph Equation Graph

13

2y x

( ) 3 2g x x

3. Write an equation for each graph. Equation Graph Equation Graph

x 0 1 2 3 4

y 180 160 140 120 100

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRxqFQoTCPbrhr22gMcCFQdsHgodu2UCBw&url=http://www.sparknotes.com/math/algebra1/graphingequations/section5.rhtml&ei=WNK4VfbfKYfYebvLiTg&bvm=bv.98717601,d.dmo&psig=AFQjCNHDxnd_mp9HyHB10aOaAASQ63n8Pg&ust=1438262221369839

5



APPLICATIONS

4. Which of the following would model the distance, D, a driver is from Chicago if they are heading towards

the city at 58 miles per hour and started 256 miles away?

(1) 256 58D t (3) 58 256D t

(2) 256 58D t (4) 58 256D t

5. The cost, C, of producing x-bikes is given by 22 132C x . The revenue gained from selling x-bikes is given

by 350R x . If the profit, P, is defined as P R C , then which of the following is an equation for P in

terms of x?

(1) 328 132P x (3) 328 132P x

(2) 372 132P x (4) 372 132P x

6. The average temperature of the planet is expected to rise at an average rate of 0.04 degrees Celsius per year

due to global warming. The average temperature in the year 2000 was 14.71 degrees Celsius.

(a) Write a function to represent the average

temperature of the planet, C(x), where x

represents the number of years since 2000.

(b) Algebraically determine the number of

years, x, it will take for the temperature, C,

to reach 20 degrees Celsius. Round to the

nearest year.

(c) Sketch a graph of the average yearly

temperature below for the interval

. Be sure to label your y-axis

scale as well as two points on the line (the y-

intercept and one additional point).

(d) What does this model project to be the

average global temperature in 2200?

6



LESSON #60 - EXPONENTIAL GROWTH AND DECAY


Last lesson, we looked at situations that can be modeled by linear functions because the values increased or

decreased by equal differences over equal intervals.

There are many things in the real world that grow faster as they grow larger or decrease slower as they get smaller.

One specific example of this is situations where the increase or decrease by equal factors over equal intervals.

These types of phenomena are known as exponential growth and decay, respectively.

Exercise #1: Last lesson, you learned the most common way to write the equation of a line is in the form,

___________.

What information can be determined by an equation in this form?

Exercise #2: Last unit, you were introduced to exponential functions in the form ___________.

What information can be determined by an equation in this form?

The value of a is the _________________.

The value of b is the ___________________.

Exercise #3: Write a function to model each of the patterns in the tables below.

x 0 1 2 3 4

y 2 6 18 54 162

x 2 1 0 1 2

y 32 16 8 4 2

7



Exercise #4: The number of people who have heard a rumor often grows exponentially. Consider a rumor that

starts with 3 people and where the number of people who have heard it doubles each day that it spreads.

(c)We’d like to determine the number of people who know the rumor after 20 days, but to do that, we need to

develop a formula to predict N (the number knowing the rumor) if we know d (the number of days it has been

spreading). Write a function to model this situation.

(d) Graph N over the interval ______________.

(e) How many people would know the rumor after

20 days?

d 0 1 2 3 4 5

N 3 6

(a) Why does it make sense that the number of

people who have heard a rumor would grow

exponentially?

(b) Fill in the table below for the number of people,

N, who knew the rumor after it has spread a

certain number of days, d.

(f) Exponential growth can be very fast. Assuming

our equation from (b) holds, how many days will

it take for the number of people knowing the

rumor to surpass the population of the United

States, which is approximately 315 million

people?

8



Let’s now look at developing a fairly simple exponential decay problem.

Exercise #5: Helmut (from Finland) is heading towards a lighthouse in a very peculiar

way. He starts 160 feet from the lighthouse. On his first trip he walks half the distance

to the light house. On his next trip he walks half of what is left. On each consecutive

trip he walks half of the distance he has left. We are going to model the distance, D,

that Helmut has remaining to the lighthouse after n-trips.

Remember, when things get crazy next unit, those exponential functions still model situations where there the

growth or decay shows a multiplying pattern. (equal factors over equal intervals). When the formulas get crazy,

any formula* can be rewritten in the form _________ to identify the start value or y-intercept, a, and growth

factor, b.

*(The only exception are exponential formulas with a vertical shift).

160 ft

(a) Fill in the table below for the amount of distance

that Helmut has left after n-trips. (b) Each entry in the table could be found by

multiplying the previous by what number? This

is important because we always want to think

about exponential functions in terms of

multiplying.

(c) Exponential decay formula:

n 0 1 2 3 4

D (ft) 160 80

(e) How far is Helmut from the windmill after 6

trips? Provide a calculation that justifies your

answer and don’t forget those units!

(f) Helmut believes he will reach the windmill after

10 trips. Is he correct? (g) Explain why Helmut will never reach the

windmill?

9



LESSON #60 - EXPONENTIAL GROWTH AND DECAY


FLUENCY

1. Write a function to model each of the patterns in the tables below.

a) x 0 1 2 3 4

y 100 50 25 12.5 6.25

b)

x 0 1 2 3 4

y 4 6 9 13.5 20.25

APPLICATIONS

2. A typical cell phone is 5 ounces (oz.). When a cell phone is thrown in the garbage and decomposes over time,

half of it is absorbed into the ground every year.

(a) Fill in the table below for the ounces of remaining cell phone, Z,

remaining after a certain number of years, y.

y 0 1 2 3 4

Z 5 2.5

(c) Using your formula from (b), how many ounces will be left after 15 years?

(b) Using the table, determine an exponential decay formula for the number of ounces remaining, Z, after

y years.

10



3. 5 students at our school have a stomach virus. The number of students infected with the virus triples each

day.

(a) Fill in the table below for the number of people, N,

that have the virus after a certain number of days, d.

d 0 1 2 3 4

N 5 15

(c) Graph N on the grid to the right over the interval 0 5d .

(d) Using your formula from (b), how many students will be infected with the stomach virus by day 8?

(e) Is your answer in part (c) reasonable? Explain.

(b) Using the table, determine an exponential growth formula for the number of students infected, N, after

d days.

11



LESSON #61 - LINEAR VERSUS EXPONENTIAL


Linear and exponential functions share many characteristics. This is because they are based on two different, but

similar, sets of principles.

Exercise #1: The two tables below represent a linear function and an exponential function. Determine which

function is linear and which function is exponential and write a function rule for each.

Exercise #2: There were 50 squirrels at a park initially. It has been noted that each year, the number of squirrels

grows by 20%.

Fill out the table below. Round the number of squirrels to the nearest whole number.

a)

Year

t

Number of

Squirrels

S(t)

0 50

1

2

3

4

5

b) Based on the table to the right, are functions

that grow by a constant percent linear or

exponential functions?

c) A function, S(t) to model the number of squirrels after t years.

LINEAR VERSUS EXPONENTIAL

Linear functions are based on repeatedly adding the same amount (the slope).

Exponential functions are based on repeatedly multiplying by the same amount (the base).

TABLE 1

x 0 1 2 3 4

y 5 10 20 40 80

TABLE 2

x 0 1 2 3 4

y 8 11 14 17 20

12



You will learn more about this next unit, but a situation where the data increases or decreases by a constant

percent can be modeled by an exponential function.

Exercise #3: Answer the following questions about linear vs. exponential functions.

a) Which situation could be modeled with an exponential function?

(1) The amount of money in a savings account where $150 is deducted every month

(2) The amount of money in Suzy’s piggy bank which she adds $10 to each week

(3) The amount of money in a certificate of deposit that gets 4% interest each year

(4) The amount of money in Jaclyn’s wallet which increases and decreases by a

different amount each week

b) Which statement below is true about a linear function?

(1) Linear functions grow by equal factors over equal intervals.

(2) Linear functions grow by equal differences over equal intervals.

(3) Linear functions grow by equal differences over unequal intervals.

(4) Linear functions grow by unequal factors over equal intervals.

c) The selling prices for a group of cars were recorded when the cars were new and for an additional five

years. The results are summarized in the tables below. Which car’s price dropped at a constant percent

rate each year?

d) Joseph’s taxi charges $10.00 for the initial service of any drive. The fee for each mile of the taxi ride is

$0.75. Which type of function is represented by this situation?

(1) Linear (2) Exponential

(3) Quadratic (4) Absolute Value

13



We want to be very sure that we understand the various constants or parameters that come up in linear and

exponential functions. Because these parameters always have a meaning in a physical situation.

4. eMathInstruction is keeping track of the number of views on a newly released math lesson screencast. They

record the total number of views as a function of the number of days since it launched, with the launch

day being 0x . The data does not follow a perfect linear or exponential pattern, so they found both the best

linear and exponential models for the data.

68 157y x 56 1.18x

y

(c) Why is the interpretation of the 157 in the linear model unreasonable or nonviable?

Exercise #3: A situation modeled by a linear function has a starting value of 1000 and increases by 100 each

day. A situation modeled by an exponential function has a starting value of 10 and doubles each day.

(a) Write a function, L(x), to model the linear function after x days.

(b) Write a function, E(x), to model the exponential function after x days.

The graph above illustrates an important point about linear and exponential functions. Even if the linear

function has a large starting value and a very steep slope, an increasing exponential function will eventually

have a larger function value because its slope is always increasing.

(a) How can you interpret the parameter 68 in the

linear model in terms of the views of the

website?

(b) How do you interpret the parameter 1.18 in the

exponential model in terms of the views of the

website?

(c) Use your graphing calculator to graph and label

L(x) and E(x). Use an appropriate viewing

window to find the intersection of the two

functions to the nearest tenth.

14



LESSON #61 - LINEAR VERSUS EXPONENTIAL


FLUENCY

1. For each of the following problems a table of values is given where 1x . For each, first determine if the

table represents a linear function, of the form y mx b , or an exponential function, of the form x

y a b .

Then, write its equation.

(a) (b)

2. The data shown in the table below represents either a linear or an exponential function. Which of the equations

below best models this data set?

(1) 5 2x

y (3) 2 10y x

(2) 10 2x

y (4) 10 5y x

3. Answer the following questions about linear vs. exponential functions.

a) Which situation could be modeled by a linear function?

(1) The height of a ball that is thrown in the air

(2) The price of a car that depreciates 20% per year

(3) The amount of money Jonathan pays for a certain number of gallons of gas at $3.85

per gallon

(4) A bacteria colony which doubles in number every 4 hours

b) Joseph conducted a science experiment involving the growth of bacteria. He measured the number of

bacteria hourly for 6 hours. The data is summarized in the accompanying table. What type of

regression would best fit the data?

(1) Linear

(2) Exponential

(3) Quadratic

(4) Absolute

Value

x -1 0 1 2 3

y 20 15 10 5 0

Type: ____________________________

Equation: _________________________

x 0 1 2 3 4

y 2 20 200 2000 20,000

Type: ____________________________

Equation: _________________________

x 1 2 3 4

y 10 20 40 80

Hours Number of Bacteria

0 250

1 750

2 2250

3 6750

15



c) The tables below show the amount of money in different bank customer’s accounts on the first day of

each month for five months. Which customer’s account increased at a constant numerical rate each

month?

4. Two scenarios are modeled using in (a) a linear function and in (b) an exponential function. In each case

interpret the parameters that help define the functions.

5. In the lesson and in the question above, we saw exponential functions with a constant percent increase. The

following function models a situation where there is a constant percent decrease. ( ) 5(.75)xf x . By what

percent is this function decreasing?

(a) Plant managers at a local tire factory model the

cost, c, in dollars of producing n-tires in a day by

the equation:

6.50 1,245c n n

Interpret the parameter values of 6.50 and 1,245.

Include units in your answer.

(b) Biologists model the population, p, of lactic acid

bacteria in yogurt as a function of the number of

minutes, m, since they added the bacteria using

the equation:

135 1.28m

p m

Interpret the parameter values of 135 and 1.28.

Include units in your answer.

16



LESSON #62 - WRITING EQUATIONS OF LINEAR AND EXPONENTIAL FUNCTIONS FROM TWO POINTS


For both linear and exponential functions only two points are necessary to determine the equation of the curve.

Exercise #1: Consider the two points 0,12 and 1, 3 . Create a linear equation that passes through these points

in y mx b form and an exponential equation in x

y a b form that also passes through them. Then, using

your calculator, graph both using a WINDOW of 2 2x and 5 15y .

Linear:

m = _____

b = _____

Equation:

Exponential:

a = _____

b = _____

Equation:

The situation above was pretty simple because the y-intercept of the function was given as well as the point where

x=1. The equation of any linear or exponential function can be found if there are two points, even if those points

do not include the y-intercept or consecutive integer x-values.

This is possible because these two points can create a system of two equations to solve for the two parameters

in the function that are needed. For linear functions these parameters are m, the slope, and b, the y-intercept. For

exponential functions, these parameters are a, the y-intercept, and b, the growth factor.

Exercise #2: Find a linear equation that passes through the points 2, 36 and 5,121.5 .

Steps

1. Plug each point into the equation, y=mx+b to

create a system of two equations.

2. Solve for b in one of the equations.

3. Solve for m using substitution.

4. Substitute the value of m into one of the two

equation to solve for b.

5. Plug the values of m and b into y=mx+b.

y

x

17



Exercise #3: Find an exponential function that passes through the points 2, 36 and 5,121.5 .

Steps

1. Plug each point into the equation, ( )xy a b to

create a system of two equations.

2. Divide one of the functions by the other to

eliminate a.

3. Solve for b.

4. Substitute the value of b into one of the two

equations to solve for a.

5. Plug the values of a and b into ( )xy a b .

Write an exponential and a linear function that pass through the points (4,98) and (9,189) .

Linear: Exponential (round values to the nearest hundredth):

In the above problem, what additional information would you need to help you determine if an exponential

function or a linear function or something else would best model this situation? There are multiple answers to

this question.

18



Exercise #4: Find the equation of the exponential function shown graphed below. Be careful in terms of your

exponent manipulation. State your final answer in the form x

y a b .

y

x

19



LESSON #62 - WRITING EQUATIONS OF LINEAR AND EXPONENTIAL FUNCTIONS FROM TWO POINTS


FLUENCY

1. Consider the points 0, 5 and 1,15 .

2. Find the equation of the linear function and the exponential function that passes through

2,192 and 5,12288 . Show the work that you use to arrive at your answer.

Linear: Exponential:

(a) Write the equation of the line that passes

between these two points in y mx b form.

(b) Write the equation of the exponential that

passes between these two points in x

y a b

form.

(c) Using your calculator, sketch the two curves on

the axes below. Label with their equations.

3

30

20



3. Find the equation for an exponential function that passes through the points 2,14 and 7, 205 in x

y a b

form. When you find the value of b do not round your answer before you find a. Then, find both to the nearest

hundredth and give the final equation.

4. Engineers are draining a water reservoir until its depth is only 10 feet. The depth decreases exponentially as

shown in the graph below. The engineers measure the depth after 1 hour to be 64 feet and after 4 hours to be

28 feet. The engineers found the exponential function 84.31(0.76)xy to model the depth of the water after

x hours. Graph the horizontal line 10y and find its intersection to determine the time, to the nearest tenth

of an hour, when the reservoir will reach a depth of 10 feet.

Wa

ter D

epth

(ft

)

Time (hrs)

21



LESSON #63 – USING REGRESSION TO FIND EQUATIONS OF FUNCTIONS


In this lesson we will look at a number of different types of problems we have done throughout the year where

we found equations of functions given sets of points. Each of the problems used a different algebraic method

because a different type of function is being found..

We will be looking at linear, exponential, quadratic, and cubic problems.

First, we will review how to find a regression equation which you learned in CC Algebra.

Steps

1) Press STAT

2) Choose #1: Edit

3) Enter the two lists in L1 and L2.

4) Press STAT. Move over to Calc.

5) Choose one of the following:

4: LinReg (ax+b) 5: QuadReg 6: CubicReg 0:ExpReg

6) Press ENTER through the options in the menu.

7) The equation set-up will be given to you in the first line. Plug in the values of a,b,c, and d. (Some types of

functions do not have all of these values.)

Exercise #1: Using the table below, find the following regression equations.

a) Find the linear regression equation that passes through the points in the

table. Round coefficients to the nearest hundredth.

b) Find the exponential regression equation for the points in the table. Round

coefficients to the nearest hundredth.

c) Find the quadratic regression for the points in the table. Round

coefficients to the nearest hundredth.

d) Find the cubic regression for the points in the table. Round coefficients to the nearest hundredth.

Each of the regression equations in exercise #1 gives the function of BEST FIT (linear, exponential, quadratic

or cubic depending on what is chosen). That means the function does not necessarily pass through all of the

points in the table, but it comes as close as possible to the points for that type of function.

In each of the following problems, we are interested in finding the EXACT equation of a certain type of

function with the given qualities.

x y

1 6

2 9.5

3 13

4 15

5 16.5

6 17.5

7 18.5

8 19

9 19.5

10 19.7

11 19.8

22



Exercise #2: Find the equation of a line passes through the points 5, 2 and 20, 4 .

a) Traditional Work:

b) With a regression:

Type:

Equation:

Exercise #3: Find the equation of the exponential function, in the form x

y a b that pases through the points

2,192 and 5,12288 .



Type:

Equation:

23



Exercise #4: Create the equation of a quadratic polynomial, in standard form, that has zeroes of 5 and 2 and

which passes through the point 3, 24 . Sketch the graph of the quadratic below to verify your result.



Type:

Equation:

Exercise #5: Create an equation for a cubic function, in standard form, that has x-intercepts given by the set

3,1, 7 and which passes through the point 2, 54 . Sketch your result on the axes shown below.



Type:

Equation:

24



LESSON #63 - USING REGRESSION TO FIND EQUATIONS OF FUNCTIONS


Directions: For problems 1-5, use regression to find the type of function that fits the given criteria.

1. Find the equation of the exponential function, in the form x

y a b that passes through the points

2, 45( ) and 4, 405( ).

Type:

Equation:

2. Find the equation of a line passes through the points 2, 45( ) and 4, 405( ).

Type:

Equation:

3. Create an equation for a cubic function, in standard form, that has x-intercepts given by the set

-8,-3,1{ } and which passes through the point

-4, - 40( ) .

Type:

Equation:

4. Create the equation of a quadratic polynomial, in standard form, that has zeroes of -1 and 3 and which

passes through the point 6,126( ).

Type:

Equation:

25



5. In problems 1 and 2, you found the equation of an exponential function and a linear function that passes

through the points 2, 45( ) and 4, 405( ).

a) What happens when you try to find the equation of a quadratic function that passes through those two

points?

b) Why do you think this happens? (Hint: Compare the information you were given in this problem to the

other two quadratic problems on the previous page).

6. In the next lesson we will be looking for exponential and linear patterns in data. Determine if each of

the following scatter plots shows an exponential pattern or a linear pattern.

26



LESSON #64 – LINEAR AND EXPONENTIAL REGRESSION


Real life data often forms patterns that can be modeled with functions. For this course, we will be looking at

data that can be modeled by linear and exponential functions as well as trigonometric functions next unit.

Below is a summary of what to look for when deciding if data should be represented by a linear function or an

exponential function.

Exercise #1: Which type of function (linear or exponential) would best model the data in each scatter plot?

Type linear y = ax + b

exponential y = abx

Connection y=2x-7

y=-4x+3

y = 2x

y = 100(.7)x

Examples

Qualities Does the plotted data resemble a straight

line?

The slope may be either positive or

negative.

Does the plotted data appear to grow (or

decline) by percentage increases (decreases)?

Remember the shape of the exponential

function.

The range must be: y > 0

27



Creating a Scatter Plot

Go to STAT #1 (Edit).

Enter the two lists in L1 and L2.

Press Y=

Move your cursor up to PLOT 1 and press ENTER to turn it on. It should be highlighted.

Press ZOOM 9 (ZoomStat) Label the window

Exercise #2: Create a scatter plot for the data. Based on the scatter plot, what type of function would best

model this data?

In each of the following problems, bivariate (two variable) data was collected which can be modeled by either a

linear or an exponential function.

Exercise #3: A scientist in a laboratory collected data about the number of bacteria in a sample. The results are

recorded in the table below.

a) Based on a scatter plot, what type of regression would best model the data (see above)?

b) Find the appropriate regression equation with coefficients rounded to the

nearest hundredth.

c) Using your regression, how many bacteria will be in the sample 4.5 hours

after the observation began? Round to the nearest whole number.

d) Using a graph of your regression, after how many hours, to the nearest

tenth of an hour, will there be 2000 bacteria in the sample?

(You do not know how to solve this problem algebraically, yet).

Hours since

observation

began

# of bacteria in the sample

0 20

1 40

2 75

3 150

4 297

5 510

Hours since

observation

began

# of bacteria

in the sample

0 20

1 40

2 75

3 150

4 297

5 510

28



The following problem has variables defined in a way that makes the equation less messy. The trade-off is that

you will have to think about these definitions when you are making predictions.

Exercise #4: The availability of leaded gasoline in New York State is decreasing, as shown in the accompanying

table where x is defined as years after 2000.

Year 2000 2004 2008 2012 2016

Gallons Available

(in thousands)

150 124 104 76 50

a) Based on a scatter plot, what type of regression would best model the

data?


nearest tenth.

c) Using your regression, how many gallons will be left in 2018 to the nearest gallon?

d) If this relationship continues, during what year will leaded gasoline first become unavailable in New York

State?

e) Using your regression equation, when will 25,000 gallons of gasoline be left in New York State?

29



Exercise #5: The data at the below shows the cooling temperatures of a freshly brewed cup of coffee after it is

poured. The brewing pot temperature is approximately 180º F.

a) Based on a scatter plot, what type of regression

would best model the data?

b) Find the appropriate regression equation with

coefficients rounded to the nearest hundredth.

c) Using the regression, to the nearest minute, when is the coffee at a temperature of 106 degrees?

d) Using the regression equation, what is the predicted temperature of the coffee after 1 hour to the nearest

degree?

e) In 1992, a woman sued McDonald's for serving coffee at a temperature of 180º that caused her to be

severely burned when the coffee spilled. As a result of this famous case, many restaurants now serve coffee

at a temperature around 155º. Using this regression, how many minutes should restaurants wait before

serving coffee, to ensure that the coffee is not hotter than 155º? Round to the nearest minute.

Time

(mins)

Temp

(º F)

0 179.5

5 168.7

11 149.2

15 141.7

18 134.6

25 123.5

30 116.3

38 109.1

42 105.7

50 100.5

30



LESSON #64 - LINEAR AND EXPONENTIAL REGRESSION


1. Which of the following scatter plots shows data best modeled by an exponential equation?

(1) (2) (3) (4)

2. The total fat and the number of calories in various McDonalds sandwiches are recorded below.

a) Based on a scatter plot, what

type of regression would best model

the data?

b) Find the appropriate

regression equation with coefficients rounded to the nearest

hundredth.

c) Using your regression, how much fat was is in a sandwich that has 700 calories? Round to the nearest tenth

of a gram.

d) Use your regression to predict the total calories in a sandwich that has 25 grams of fat. Round to the nearest

calorie.

Sandwich

Total

Fat

(g)

Total

Calories

Hamburger 9 260

Cheeseburger 13 320

¼ pounder

(QP) 21 420

QP with

cheese 30 530

Big Mac 31 560

Fish Fillet 28 500

y

x

y

x

y

x

y

x

31



3. The table below, created in 1996, shows a history of transit fares from 1975 to 2010.

Note: x is measured in years after 1950.

Year 20 25 30 35 40 45 50 55 60

Fare ($) 0.10 0.15 0.20 0.30 0.40 0.60 0.80 1.15 1.50

a) Based on a scatter plot, what type of regression would best model the data?


nearest thousandth.

c) Using your regression, predict when the transit fare will reach $3.00 to the nearest year.

d) Using your regression, what will the fare be in the year 2017 to the nearest cent?

e) Using your regression, predict when the transit fare will reach $6.00 to the nearest year.

f) It took a long time for the price of the fare to reach $3.00. Why was there much less time between when the

fare reached $3.00 and when it reached $6.00?

ESSON #59 LINEAR FUNCTIONS AND MODELING ......Last lesson, we looked at situations that can be modeled by linear functions because the values increased or decreased by equal differences

Documents