Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.4, Slide 1 Chapter 2 Modeling with Linear Functions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.4, Slide 1

Chapter 2Modeling with Linear Functions


2.4 Slope Is a Rate of Change


Ratio and Unit Ratio

The ratio of a to b is the fraction

A unit ratio is a ratio written as with b = 1.

.ab

ab


Formula for Rate of Change and Average Rate of Change

Suppose a quantity y changes steadily from y1 to y2 as a quantity 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:

If either quantity changes, but not steadily, then this formula is the average rate of change of y with respect to x.

2 1

2 1

change in change in

y y yx x x


Example: Finding Rates of Change

1. The number of Americans who fly to Europe has decreased approximately steadily from 13.3 million in 2007 to 10.8 million fliers in 2011 (Source: U.S. Department of Commerce). Find the average rate of change of the number of Americans who flew to Europe.


Example: Finding Rates of Change

2. In Phoenix, Arizona, the average value of a two-bedroom home is $129 thousand and the average value of a four-bedroom home is $338 thousand. (Source: Trulia). Find the average rate of change of the average value of a home with respect to the number of bedrooms.


Solution

1. The average rate of change of the annual number of Americans who flew to Europe was about –0.63 million fliers per year. So, on average, the number of Americans who flew to Europe declines yearly by about 0.63 million (630 thousand) fliers.

change in the

number of fliers 10.8 million fliers 13.3 million flierschange in time year 2011 year 2007

2.5 million fliers 0.63 million fliers4 year 1 year


Solution

2. To be consistent in finding the signs of the changes, assume that the number of bedrooms increases from two to four and that the average value increases from $129 thousand to $338 thousand.


Solution

2. The average rate of change of the average value with respect to the number of bedrooms is $104.5 thousand per bedroom. So, the average value increases by $104.5 thousand per bedroom.

change in

average value $338 thousand $129 thousandchange in 4 bedrooms 2 bedrooms

number of bedrooms

$209 thousand $104.5 thousand2 bedrooms 1 bedroom


Increasing and Decreasing Quantities

Suppose 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.


Example: Comparing Slope with a

Rate of Change

Suppose a student drives at a constant rate. Let d be the distance (in miles) the student can drive in t hours. Some values of t and d are shown in the table at the right.


Example: Comparing Slope with a

Rate of Change

1. Create a scattergram. Then draw a linear model.2. Find the slope of the linear model.3. Find the rate of change of the distance traveled

for each given period. Compare each result with the slope of the linear model.a. From t = 2 to t = 3b. From t = 0 to t = 4


Solution

1. Draw the scattergram and then draw a line that contains the data points.


Solution

2. The slope formula is So, with the2 1

2 1

.y y

mx x

variables t and d, we have

180 1203 1

02

660

Arbitrarily use the points (2, 120) and (3, 180) to calculate the slope:

So, the slope is 60.

2 1

2 1

.d dt t


Solution

3. a. Calculate the rate of change of the distance traveled from t = 2 to t = 3:

change in distance 180 miles 120 mileschange in time 3 hours 2 hours

60 miles60 miles per hour

1 hour

The rate of change (60 miles per hour) is equal to the slope (60).


Solution

3. b. Calculate the rate of change of the distance traveled from t = 0 to t = 4:

change in distance 240 miles 0 mileschange in time 4 hours 0 hours

240 miles 60 miles60 miles per hour

4 hours 1 hour

The rate of change (60 miles per hour) is equal to the slope (60).


Slope Is a Rate of Change

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.


Constant Rate of Change

Suppose 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 linear relationship between t and p.


Unit Analysis

Definition

We perform a unit analysis of a model’s equation by determining the units of the expressions on both sides of the equation. The simplified units of the expressions on both sides of the equation should be the same.


Example: Finding a Model

A driver fills her car’s 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.


Example: Finding a Model

1. Is there a linear relationship between d and G? Explain.

2. Find the G-intercept of a linear model.3. Find the slope of the linear model.4. Find an equation of the linear model.5. Perform a unit analysis of the equation.


Solution

1. Since the rate of change of gallons remaining with respect to distance traveled is a constant (–0.04 gallons per mile), the variable d and G are linearly related.

2. When the tank was filled, it contained 12 gallons of gasoline. We can represent this by the ordered pair (0, 12), which is the G-intercept.


Solution

3. The rate of change of gasoline remaining in the tank with respect to distance traveled is –0.04 per mile. So, the slope is –0.04.

4. An equation of the linear model can be written in the form G = md + b. Since slope is –0.04 and the G-intercept is (0, 12) we have G = –0.04d + 12.


Solution

5.

So, the units on both sides of the equation are gallons, which suggests the equation is correct.


Example: Analyzing a Model

Sales of smartphones are shown in the table for various years. Let s be smartphone annual sales (in millions of phones) at t years since 2000. A model of the situation is s = 7.8t – 33.5.


Example: Analyzing a Model

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 points.

2. What is the slope of the model? What does it mean in this situation?

3. Find the rates of change of sales from one year to the next. Compare the rates of change with the result in Problem 2.

4. Predict the sales in 2018.


Solution

1. Draw a scattergram and the model in the same viewing window. The line comes close to the data points, so the model is a reasonable one.


Solution

2. The slope is 7.8, because s = 7.8t – 33.5 is of the form y = mx + b and m = 7.8. According to the model, sales are increasing by 7.8 million smartphones per year.


Solution

3. The rates of change of sales are shown in the table below. All of the rates of change are fairly close to 7.8 million smartphones per year.


Solution

4. Substitute the input 18 for t in the equation.

s = 7.8(18) – 33.5 = 106.9

According to the model, smartphone sales will be 106.9 million phones in 2018.


Slope Is an Average Rate of Change

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.


Describing the Meaning of Slope

WarningIt is a common error to be vague in describing the meaning of the slope of a model. A description such as “The slope means it is increasing.” neither specifies the quantity that is increasing nor the rate of increase.

The following statement includes the missing information.“The slope of 7.8 means sales increased by 7.8 million smartphones per year.”

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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.4, Slide 1 Chapter 2 Modeling with Linear Functions