Section 2.4
Slope Is a Rate of Change
Definition
Calculate the Rate of Change
Definition
a
The ratio of a to b is the fraction . A unit ratio
b
a
is a ratio written as with b  1.
b
Example
Suppose the sea level increases steadily by 12 inches
in the past 4 hours as it approaches high tide. We can
compute how much sea level change per hour by
finding the unit ratio of the change in sea level to the
change in time:
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 2
Definition
Calculate the Rate of Change
Solution
12 inches 3 inches

4 hours
1 hours
So, sea level increases by 3 inches per hours.
• This is an example of rate of change
• We say, “The rate of change of sea level with
respect to time is 3 inches per hour.”
• The rate of change is constant because sea level
increases steadily
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 3
Examples of Rates of Change
Calculate the Rate of Change
Examples
Examples of rates of changes:
• The number of members of a club increases by five
people per month.
• The value of a stock decreases by $2 per week.
• The cost of a gallon of gasoline increases by 10¢
per month.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 4
Formula for Rate of Change and Average Rate of Change
Calculate the Rate of Change
Definition
Suppose that a quality y changes steadily form y1 to
y2 as a quality x changes steadily from x1 to x2. Then
the rate of change of y with respect to x is the ratio
of the change in y to the change in x:
change in y y2  y1

change in x x2  x1
If either quantity does not change steadily, then this
formula is the average rate of change of y with
respect to x.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 5
Finding Rates of Change
Calculate the Rate of Change
Example
1. The number of fires in U.S. hotels declined
approximately steadily from 7100 fires in 1990 to
4200 in 2002. Find the average rate of change of
the number of hotel fires per year between 1990
and 2002.
Solution
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 6
Finding Rates of Change
Calculate the Rate of Change
Solution Continued
The average rate of change of the number of fires per
year was about –241.67 fires per year. So, on
average, the number of fires declined yearly by about
242 fires.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 7
Finding Rates of Change
Calculate the Rate of Change
Example Continued
2. In San Bruno, California, the average value of a
two-bedroom home is $543 thousand, and the
average value of a five-bedroom home is $793.
Find the average rate of change of the average
value of a home with respect to the number of
bedrooms.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 8
Finding Rates of Change
Calculate the Rate of Change
Solution
• Consistent in finding signs of the changes
• Assume that number of bedrooms increases form
two to five
• Assume that the average value increases from $543
thousand to $793 thousand
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 9
Finding Rates of Change
Calculate the Rate of Change
Solution Continued
• Average rate of change of the average value with
respect to the number of bedrooms is about $83.33
thousand per bedroom
• Average value increases by about $83.33 thousand
per bedroom
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 10
Increasing and Decreasing Quantities
Calculate the Rate of Change
Properties
Suppose that a quantity p depends on a quantity t:
• If p increases steadily as t increases steadily, then
the rate of change of p with respect to t is positive
• If p decreases steadily as t increases steadily, then
the rate of change of p with respect to t is negative
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 11
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Example
Suppose that 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.
Then draw a linear model.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 12
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution
• Draw a scattergraph
that contains the points
Example Continued
2. Find the slope of the
linear model.
Solution
y2  y1
Slope formula is m 
, replacing y and x with
x2  x1
d and t, respectively, we have:
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 13
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
d 2  d1
m
t2  t1
Arbitrarily use the points (2, 120) and (3, 180) to
calculate the slope:
180  120 60
m
  60
3 2
1
• The slope is 60
• Checks with calculations shown in the scattergraph
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 14
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Example Continued
3. Find the rate of change of distance per hour for
each given period. Compare each result with the
slope of the linear model.
a. From t  0 to t=3
b. From t  0 to t=4
Solution
• Calculate rate of change of the distance per hour
from t  0 to t=3
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 15
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
• The rate of change (60 miles per hour) is equal to
the slope (60)
• For part b., calculate the rate of change of distance
per hour from
t  0 to t=4
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 16
Comparing Slope with a Rate of Change
Slope Is a Rate of Change
Solution Continued
• The rate of change (60 miles per hour) is equal to
the slope (60)
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 17
Slope is a Rate of Change
Slope Is a Rate of Change
Property
If there is a linear relationship between quantities t
and p, and if p depends on t, then the slope of the
linear model is equal to the rate of change of p with
respect to t.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 18
Constant Rate of Change
Slope Is a Rate of Change
Property
Suppose that a quantity p depends on a quantity t:
• If there is a linear relationship between t and p, then
the rate of change of p with respect to t is constant.
• If the rate of change of p with respect to t is
constant, then there is a liner relationship between t
and p.
Section 2.4
Lehmann, Intermediate Algebra, 3ed
Slide 19
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, 3ed
Slide 20
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, 3ed
Slide 21
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, 3ed
Slide 22
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, 3ed
Slide 23
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, 3ed
Slide 24
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, 3ed
Slide 25
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, 3ed
Slide 26
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
linear model is: G  0.04t  12
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, 3ed
Slide 27
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, 3ed
Slide 28
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, 3ed
Slide 29
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, 3ed
Slide 30
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
• s  0.17t  2.15 which is of the form y  mx  b
• 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, 3ed
Slide 31
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, 3ed
Slide 32
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, 3ed
Slide 33