2.1 Using Lines to Model Data (Page 1 of 21)
2.1 Using Lines to Model Data
Scattergram and Linear Models
The graph of plotted data pairs is called a scattergram. A linear
model is a straight line or an equation that describes the
relationship between two quantities for a true-to-life situation.
Example 1
Let v represent the number of
visitors (in millions) to the
Grand Canyon in the year that
is t years since 1960.
a. Fill-in the table of values
for t. Identify the
dependent and
independent variables.
Years
Number of Visitors
Year since 1960
(millions)
t
v
1960
1.2
1970
2.3
1980
2.6
1990
3.8
2000
4.8
b. Make a scattergram of the data.
c. Use a ruler to draw (“eyeball”) a
line (linear model) that fits the
data well. As always, label and
scale both axes.
d. Use the linear model to estimate
the number of visitors in 2010
(extrapolation).
e. Use the linear model to estimate
in what year there will be 4 million visitors to the Grand
Canyon (interpolation).
2.1 Using Lines to Model Data (Page 2 of 21)
Interpolation, Extrapolation & Model Breakdown
Interpolation is making prediction within the data given.
Extrapolation is making a prediction outside the data given. When
a model yields a prediction that does not make sense or an estimate
that is not a good approximation, we say that model breakdown
has occurred. Model breakdown mostly occurs when trying to
make an estimate outside the range given in the data (called
extrapolation).
2.1 Using Lines to Model Data (Page 3 of 21)
Example 2
Prozac is an antidepressant that
was approved by the FDA in
1987. Let p be the number of
prescriptions of Prozac (in
millions) dispensed at t years
since 1980.
a. Fill-in the table of values for
t.
Year
1989
1991
1993
1995
1996
Number of
t Prozac Prescriptions
(millions)
6.1
10.0
12.2
18.8
20.7
b. Identify the dependent and independent variables.
c. Make a scattergram of the data.
d. Use a ruler to draw (“eyeball”)
a line (linear model) that fits
the data well. As always, label
and scale both axes.
e. Use the linear model to
estimate the number of Prozac
prescriptions in 1994.
f.
Use the linear model to predict when the number of Prozac
prescriptions will reach 30 million.
Year
Salmon
Population
(millions)
2.1 Using Lines to Model Data (Page 4 of 21)
Example 3
The Pacific salmon populations for various
years are listed in the table. Let P
represent the salmon population (in
millions) at t years since 1950.
a. Identify the dependent and
independent variables.
b. Make a scattergram of the data.
P
1960
1965
1970
1975
1980
1985
1990
10.02
10.00
7.61
3.15
4.59
3.11
2.22
c. Use a ruler to draw (“eyeball”) a line (linear model) that fits
the data well. As always, label and scale both axes.
d. Find the P-intercept of the model and explain its meaning.
e. Find the t-intercept and explain
its meaning.
2.1 Using Lines to Model Data (Page 5 of 21)
Example 4 (exercise 2.1 #8)
Carbon Emissions
Over the past 120 years, most scientists
Year
(billions of tons)
agree that Earth’s average temperature
1950
1.6
has increased by about 1o F. What is
1960
2.6
under debate is why this happened and
1970
3.8
whether we should be concerned. Some
4.9
scientists believe that the warming is due 1980
1990
5.9
to carbon emissions from burning fossil
fuels. Let c represent the carbon emissions at t years since 1950.
a. Identify the dependent and independent variables.
b. Make a scattergram and draw (“eyeball”) a linear model that
fits the data well. As always, label and scale the axes.
c. Use your linear model to
estimate the carbon emissions
in 1998.
d. The actual amount of carbon
emissions in 1998 was 6.3
billion tons. Explain the
difference between the actual
and estimated amounts.
2.2 Finding Regression Equations for Linear Models (Page 6 of
21)
2.2 Finding Regression Equations for Linear Models
In section 2.1 we used scattergrams and an “eyeballed best-fit” line
(linear model) to model data and make estimates. In this section
we will find the regression equations for linear models.
Linear Regression Function
The linear regression line is the line that mathematically best fits
the data. The linear regression equation, or linear regression
function, is the equation of the regression line.
Finding the Linear Regression Equation on the TI-83
1. Enter the independent variable data values into list 1 (L1) and
the corresponding dependent variable values into list 2 (L2).
To access your lists press STAT followed by ENTER.
STAT / 1:Edit
2. Press STAT PLOT followed by
ENTER. Then set the Plot1 settings
as shown. Press
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ZOOM / 9:ZoomStat
to view the scattergram.
3. Run the linear regression program:
STAT / CALC / 4:LinReg (ax+b) L1 , L2 , Y1
On the TI-83 a is the slope and (0, b) is the “y”-intercept (i.e.
the vertical axis intercept). Write a and b to three decimal
places.
3. Rewrite the equation using the variables in the application.
2.2 Finding Regression Equations for Linear Models (Page 7 of
21)
Example 1
The American life span has been
increasing over the last century. Let L
represent the life expectancy at birth for an
American born t years after 1970. Create a
scattergram of the data on your calculator
and determine if a linear model is
appropriate. If it is, then find the linear
regression model for the data.
Example 2
The Pacific salmon population for various
years are listed in the table. Let P
represent the salmon population (in
millions) at t years since 1950. Create a
scattergram of the data on your calculator
and determine if a linear model is
appropriate. If it is, then find the linear
regression model for the data.
Birth
Year
1970
1975
1977
1980
1982
1985
1987
1990
1993
1996
2000
Life
Expectancy
70.8
72.6
73.3
73.7
74.5
74.7
74.9
75.4
75.5
76.1
76.9
Number of
Years
since 1950
t
10
15
20
25
30
35
40
Salmon
Population
(millions)
P
10.02
10.00
7.61
3.15
4.59
3.11
2.22
2.3 Function Notation and Making Predictions (Page 8 of 21)
2.3 Function Notation and Making Predictions
In section 2.2 we found the linear regression equations for models.
In this section we will use the equations to make estimates and
predictions. We will also learn notation for functions.
Example 1
The table shows the average salaries for
professors at four-year colleges and
universities. Let s represent the average
salary (in thousands of dollars) at t years
since 1970.
a. Verify the linear regression equation
for s is s  1.70t  6.71 .
Year
Average Salary
(thousands of dollars)
1975
1980
1985
1990
1995
2000
16.6
22.1
31.2
41.9
49.1
57.7
b. Predict the average salary in 2008.
c. Predict when the average salary will be $75,000.
d. Explain the meaning of the slope in this situation.
2.3 Function Notation and Making Predictions (Page 9 of 21)
Example 2
In example 2 of section 2.2 we found P   0.29t  12.99 to be a
model for the salmon population (in millions) in the year that is t
years since 1950.
1a. Predict the salmon population in year 1955.
1b. Use your graphing calculator’s TABLE and TBLSET
functions to verify your prediction.
1c. Use the GRAPH and TRACE functions to verify your
prediction.
2.
a.
Predict the year when the salmon population was
6.4 million.
Algebraically
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b.
Graphically
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2.3 Function Notation and Making Predictions (Page 10 of 21)
Example 3
The average number of hours
Americans work in a week gradually
increased from 1975 to 1995. Let W
be the average number of hours
worked per week at t years since
1970.
1. Verify the linear regression
model of the data is
W  0.34t  42.36 .
Year
Number of
Years Since
1970
t
Average
hours worked
per week
W
1975
1980
1984
1989
1993
1995
2. Predict the number of hours that the average American will
work per week in the year 2003.
3. Predict when the average American will work 60 hours per
week.
4. Explain the meaning of the slope in this situation.
43.1
46.9
47.3
48.7
50.0
50.6
2.3 Function Notation and Making Predictions (Page 11 of 21)
Function Notation
When an equation is a function (as all non-vertical lines are) it is
often more convenient to use function notation. In example 2 the
regression equation is W  0.34t  42.36 , where the hours worked
per week, W, depended on the years elapsed since 1970, t. The
expression “W depends on t” is equivalently stated in function
terminology as “W is a function of t.” That is,
W  f (t)  0.34t  42.36
is read
or
or
“W equals f of t”
“W is a function of t”
“W depends on t”
where f is the name of the function and t is the independent
variable. That is,
dependent variable = f(independent variable), i.e. W  f (t)
The only difference is that in equation notation W is the dependent
variable, and in function notation f(t) is the dependent variable.
There is nothing special about naming the function f, we could
have just as easily named it r to remind us it is the regression
equation we are talking about. That is, r(t)  0.34t 18.70 .
Example 4
Let y  3x  5 and
f (x)  3x  2 .
Equation Notation
Notation
a. Find y when x  2 .
Function
b. Find f (2) .
c. Find x when y  10 .
d. Find x when f (x)  10 .
2.3 Function Notation and Making Predictions (Page 12 of 21)
2.3 Function Notation and Making Predictions (Page 13 of 21)
Example 5
Let g(x)  
1
x  3 . Find
2
a.
g(4)
b.
g(10)
c.
g(0)
d.
 2
g 
 3
e.
g(2.7)
f.
g(a)
g.
g(2a)
h.
g(a  2)
f.
x when g(x)  0 .
g.
x when g(x)  7 .
h.
x when g(x)  11 .
2.3 Function Notation and Making Predictions (Page 14 of 21)
Example 6
The graph of function g is shown. Estimate the following.
y
g(4)
1.
2.
g(0)
3.
x when g(x)  3
4.
x when g(x)  0
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x
2.3 Function Notation and Making Predictions (Page 15 of 21)
Four-Step Modeling Process
1. Create a scattergram of the data and determine if a linear model
is suited for this data.
2. Draw (“eyeball”) a line through the data to represent the linear
model. Alternatively, if asked, find the regression line for the
data.
3. Verify that the line models the data well.
4. Use the equation or graph for your model to make estimates,
make predictions, and draw conclusions.
Example 7
Smoking has been on the decline
in the United States for decades
(see table). Let p  g(t) be the
percent of Americans who
smoke at t years since 1950.
1. Verify the linear model for p
is p  0.59t  51.15 .
2. Write the regression
equation using function
notation p  g(t) .
Year
Number of
Years since
1950
Percent
Who
Smoke
1965
1974
1979
1983
1987
1992
1995
3. Find g(104) and explain its meaning.
4. Find the value for t when g(t) = 30. Explain its meaning.
5. Find the intercepts and explain their meaning.
42.4
37.1
33.5
32.1
28.8
26.5
24.7
2.3 Function Notation and Making Predictions (Page 16 of 21)
Example 8
In 1963 there were only 417 male-female pairs of
bald eagles in the United States. However, the
bald eagle has made a comeback (see table). Let
f(t) represent the number of male-female pairs of
bald eagles (in thousands) at t years since 1990.
1. Verify the linear model is suited for this data
is f (t)  0.29t  3.36 .
2. In 1999 the bald eagle was taken off the
threatened species list. Estimate the number
of bald eagle pairs that year.
Year
1993
1994
1995
1996
1997
1998
Number of
Bald Eagle
Pairs
(thousands)
4.2
4.5
4.9
5.1
5.3
5.7
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3. Find f (15) . Explain its meaning in this application.
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4. Find t when f (t)  15 . Explain its meaning in this application.
5. Explain the meaning of the slope in this situation.
2.4 Slope as a Rate of Change (Page 17 of 21)
2.4 Slope as a Rate of Change
Slope as a Rate of Change
For a linear function y  mx  b , the slope m is rate of change of
y with respect to x.
m
change in y
change in dep. variable

change in x change in indep. variable
Example 1
For each of the following, find the rate of change and explain its
meaning in the application.
a. Suppose sea level fell steadily by 12 inches in the last four
hours as the tide came in.
b.
The number of fires in U.S. Hotels declined form 7100 fires in
1990 to 4200 fires in 2002.
c.
In San Bruno, CA, the average value of a 2-bedroom home is
$543 thousand and the average value of a 5-bedroom home is
$793 thousand.
2.4 Slope as a Rate of Change (Page 18 of 21)
Example 2
Suppose a student drives at a constant rate. Let
d be the distance (in miles) that the student can
drive in t hours. Some values of t and d are
shown in the table.
1. Create a scattergram and draw a linear
model.
Time
(hours)
t
0
1
2
3
4
5
Distance
(miles)
d
0
60
120
180
240
300
2.
Find the slope of the linear model.
3.
Find the rate change of distance per hour from t = 2 to t = 3.
4.
Find the rate change of distance per hour from t = 0 to t = 4.
5.
Find the equation of the linear model.
2.4 Slope as a Rate of Change (Page 19 of 21)
Example 3
To rent a standard pickup truck, Budget Truck Rental charges a
daily fee of $37 plus $0.25 for each mile traveled. Let C represent
the cost (in dollars) of renting a pickup truck driven x miles.
1. Find an equation that models the cost.
x
0
1
2
3
4
C
2. Identify the dependent and independent variables.
3. What is the slope of the line? Explain its meaning in this
application.
Constant Rate of Change Property
If the rate of change of y with respect to x is constant, then there is
a linear relationship between the variables.
2.4 Slope as a Rate of Change (Page 20 of 21)
Example 4
A driver fills her 12-gallon gasoline tank and drives at a constant
speed. The car consumes 0.04 gallon per mile. Let G be the
number of gallons of gasoline remaining in the tank after she has
driven d miles since filling up.
a. Is there a linear relationship between d and G? Explain.
b. Find the G-intercept of the linear model and explain its
meaning in this application.
c. Find the slope of the linear model.
d. Find the equation of the linear model.
Unit Analysis
A unit analysis of a model’s equation is done by determining that
the units on both sides of the equation are the same.
e. Perform a unit analysis of the equation found in part d.
2.4 Slope as a Rate of Change (Page 21 of 21)
Example 5
Yogurt sales (in billions of dollars) in
the United States are shown in the
table. Let s by yogurt sales (billions
of dollars) in the year that is t years
since 2000.
a. Use a graphing calculator to draw
a scattergram and find the linear
model for the data.
Year
2001
2002
2003
2004
2005
Sales
(billions of dollars)
2.3
2.5
2.7
2.8
3.0
b. Explain the meaning of the slope in this situation.
c. Predict the sales in 2010.
d. Predict when the sales will be $3.5 billion.
e. Find the s-intercept and explain it meaning in this situation.
f.
Find the t-intercept and explain its meaning in this situation.