Section 2.4
Finding a Linear Model
Finding a Model
Finding an Equation of a Linear Model
Example
A company’s profit was $10 million in 2005 and has
increased by $3 million per year. Let p be the profit
(in millions of dollars) in the year that is t years since
2005.
1. Is there a linear relationship between t and p?
Solution
• Since the profit is a constant $3 million per year,
the variables t and p are linearly related
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 2
Finding a Model
Finding an Equation of a Linear Model
Example Continued
2. Find the p-intercept of a linear model
Solution
• Profit was $10 million in 2005
• 2005 is 0 years since 2005
• This gives the ordered pair (0, 10)
• So, the p-intercept is (0, 10)
Example Continued
3. Find the slope of the linear model.
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 3
Finding a Model
Finding an Equation of a Linear Model
Solution
• Rate of change of profit per year is $3 million per
year
• So, the slope of the linear model is 3
Example Continued
4. Find an equation of the linear model.
Solution
• Since p-intercept is (0, 10) and slope is 3, the linear
model is: p= 3t + 10
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 4
Finding a Model
Finding an Equation of a Linear Model
Graphing Calculator
• Verify the ordered pair (0, 10)
• Verify that as the input increases by 1, the output
increases by 3
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 5
Definition: Unit Analysis
Unit Analysis of a Linear Model
Definition
We perform a unit analysis of a model’s equations
by determining the units of the expression on both
sides of the equation. The simplified units of the
expressions on both sides of the equation should be
the same.
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 6
Finding a Model
Unit Analysis of a Linear Model
Example
A driver fills her car’s 12-gallon gasoline tank and
drives as 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.
1. Is there a linear relationship between d and G?
Solution
• Rate of change is a constant –0.04 gallons per
minute, so d and G are linearly related
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 7
Finding a Model
Unit Analysis of a Linear Model
Example Continued
2. Find the G-intercept of a linear model.
Solution
• Tank is full at 12 gallons: ordered pair (0, 12)
Example Continued
3. Find the slope of the linear model.
Solution
• Gasoline remaining in the tank with respect to
distance traveled is: m = −0.04
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 8
Finding a Model
Unit Analysis of a Linear Model
Example Continued
4. Find the equation of the linear model.
Solution
• Since p-intercept is (0, 12) and slope is –0.04, the
−0.04t + 12
linear model is: G =
Example Continued
5. Perform a unit analysis of the equation
Solution
• Here a unit analysis on the equation G =
−0.04t + 12 :
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 9
Finding a Model
Unit Analysis of a Linear Model
Solution Continued
• We use the fact that
= 1 to simplify the units of
the expression on the right-hand side of the equation:
• Units on both sides are gallons: Suggesting correct
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 10
Analyzing a Model
T w o Va r i a b l e s T h a t A r e A p p r o x i m a t e l y L i n e a r l y R e l a t e d
Example
Yogurt sales (in billions of dollars) in the United
States are shown in the table for various years.
Let s be yogurt sales (in billions of dollars) in the
year that is t years since 2000.
A model of the situation =
is: s 0.17t + 2.15
1. Use a graphing calculator to draw a scattergram
and the model in the same viewing window.
Check whether the line comes close to the data.
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 11
Analyzing a Model
T w o Va r i a b l e s T h a t A r e A p p r o x i m a t e l y L i n e a r l y R e l a t e d
Solution
• Draw in the same screen using
a graphing calculator
• See Sections B.8 and B.10
Example Continued
2. What is the slope of the
model? What does it mean in
this situation?
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 12
Analyzing a Model
T w o Va r i a b l e s T h a t A r e A p p r o x i m a t e l y L i n e a r l y R e l a t e d
Solution
y mx + b
• s 0.17t + 2.15 which is of the form=
=
• Since m is the slope, the slope is 0.17
• Sales increase by 0.17 billion dollars per year
Example Continue
3. Find the rates of change in sales from one year to
the next. Compare the rates of change with the
results in Problem 2.
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 13
Analyzing a Model
T w o Va r i a b l e s T h a t A r e A p p r o x i m a t e l y L i n e a r l y R e l a t e d
Solution
• Rates of change are shown in the table – all are
close to 0.17
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 14
Analyzing a Model
T w o Va r i a b l e s T h a t A r e A p p r o x i m a t e l y L i n e a r l y R e l a t e d
Example Continue
4. Predict the sales in 2010.
Solution
• Substitute the input of 10 for t:
Property
If two quantities t and p are approximately linearly
related, and if p depends on t, then the slope of a
reasonable linear model is approximately equal to the
average rate of change of p with respect to t.
Section 2.4
Lehmann, Intermediate Algebra, 4ed
Slide 15