Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Warm Up

1. Find the slope and y-intercept of the line that

passes through (4, 20) and (20, 24).

The population of a town is decreasing at a rate of 1.8% per year. In 1990, there were 4600 people.

2. Write an exponential decay function to model this situation.

3. Find the population in 2010.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is linear.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is quadratic.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Look at the tables and graphs below. The data show two ways you have learned that variable quantities can be related. The relationship shown is exponential.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

In the real world, people often gather data and then must decide what kind of relationship (if any) they think best describes their data.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Example 1A: Graphing Data to Choose a Model

Which kind of model best describes the data?

Time(h) Bacteria

0 24

1 96

2 384

3 1536

4 6144

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Example 1B: Graphing Data to Choose a Model

Which kind of model best describes the data?

Boxes Reams of paper

1 10

5 50

20 200

50 500

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Check It Out! Example 1a

Which kind of model best describes the data?

x y

–3 0.30

–2 0.44

0 1

1 1.5

2 2.25

3 3.38

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Check It Out! Example 1b

Which kind of model best describes the data?

x y

–3 –14

–2 –9

–1 –6

0 –5

1 –6

2 –9

3 –14

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

We’re going to use our calculators to help us find models for the following data sets

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

We’re going to use our calculators to help us find models for the following data sets

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

After deciding which model best fits the data, you can write a function. Recall the general forms of linear, quadratic, and exponential functions.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Time (s) Height (ft)

0 4

1 68

2 100

3 100

4 68

Height of golf ball

Determine which model best fits the dataand then find the equation for the model

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Determine which model best fits the dataand then find the equation for the model

Time (yr) Amount ($)

0 1000.00

1 1169.86

2 1368.67

3 1601.04

Money in CD

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Determine which model best fits the dataand then find the equation for the model

Data (1) Data (2)

–2 10

–1 1

0 –2

1 1

2 10

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Example 3: Problem-Solving Application

Use the data in the table to describe how the number of people changes. Then write a function that models the data. Use your function to predict the number of people who received the e-mail after one week.

Time (Days) Number of People Who Received the E-mail

0 8

1 56

2 392

3 2744

E-mail forwarding

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Use the data in the table to describe how the oven temperature is changing. Then write a function that models the data. Use your function to predict the temperature after 1 hour.

Check It Out! Example 3

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Lesson Quiz: Part I

Which kind of model best describes each set of data?

1. 2.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Lesson Quiz: Part II

3. Use the data in the table to describe how the amount of water is changing. Then write a function that models the data. Use your function to predict the amount of water in the pool after 3 hours.

Holt McDougal Algebra 1

9-4 Linear, Quadratic, and Exponential Models

Comparing Functions

Holt McDougal Algebra 1

9-5 Comparing Functions

Example 2: Comparing Exponential FunctionsAn investment analyst offers two different investment options for her customers. Compare the investments by finding and interpreting the average rates of change from year 0 to year 10.

Holt McDougal Algebra 1

9-5 Comparing Functions

Check It Out! Example 2

Compare the same investments’ average rates of change from year 10 to year 25.

Holt McDougal Algebra 1

9-5 Comparing Functions

Students in an engineering class were given an assignment to design a parabola-shaped bridge. Suppose Rosetta uses y = –0.01x2 + 1.1x and Marco uses the plan below. Compare the two models over the interval [0, 20].

Check It Out! Example 3

Holt McDougal Algebra 1

9-5 Comparing Functions

Example 4: Comparing Different Types of Functions

A town has approximately 500 homes. The town council is considering plans for future development. Plan A calls for an increase of 50 homes per year. Plan B calls for a 5% increase each year. Compare the plans.

Let x be the number of years. Let y be the number of homes. Write functions to model each plan

Holt McDougal Algebra 1

9-5 Comparing Functions

Two neighboring schools use different models for anticipated growth in enrollment: School A has 850 students and predicts an increase of 100 students per year. School B also has 850students, but predicts an increase of 8% per year. Compare the models.

Check It Out! Example 4

Let x be the number of students. Let y be the total enrollment. Write functions to model each school.

Holt McDougal Algebra 1

9-5 Comparing Functions

Lesson Quiz: Part III

3. A car manufacturer has 40 cars in stock. The manufacturer is considering two proposals. Proposal A recommends increasing the inventory by 12 cars per year. Proposal B recommends an 8% increase each year. Compare the proposals.