Modeling with

Linear Functions

Chapter 2

Using Lines to Model Data

Section 2.1

Lehmann, Intermediate Algebra, 3edSection 2.1

The number of Grand Canyon visitors is listed in the table for various years. Describe the data.

Slide 3

Using Lines to Model DataScattergrams

• Let v be the number (in millions) of visitors• Let t be the number of

years since 1960

Example

Solution

Lehmann, Intermediate Algebra, 3edSection 2.1

Sketch a line that comes close to (or on) the data points.

Slide 4

Using Lines to Model DataScattergrams

The graph on the left does the best job of this.

Example Continued

Lehmann, Intermediate Algebra, 3edSection 2.1

If the points in a scattergram of data lie close to (or on) a line, then we say that the relevant variables are approximately linearly related. For the Grand Canyon situation, variables t and v are approximately linearly related.

A model is a mathematical description of an authentic situation. We say that the description models the situation.

Slide 5

DefinitionsLinear Models

Definition

Lehmann, Intermediate Algebra, 3edSection 2.1

A linear model is a linear function, or its graph, that describes the relationship between two quantities for an authentic situation.

• The Grand Canyon model is a linear model• Every linear model is a linear function•Functions are used to describe situations and to describe certain mathematical relationships

Slide 6

DefinitionsLinear Models

Definition

Property

Lehmann, Intermediate Algebra, 3edSection 2.1

Use a linear model to predict the number of visitors in 2010.

Slide 7

Using a Linear Model to Make a Prediction and an Estimate

Using a Linear Model to Make Estimates and Predictions

• Year 2010 corresponds to t = 50: 2010 – 1960 = 50• Locate point on linear model for t = 50• The v-coordinate is approximately 5.6• The model estimates 5.6 million visitors in 2010

Example

Solution

Lehmann, Intermediate Algebra, 3edSection 2.1

Use a linear model to estimate the year there ware 4 million visitors.

Slide 8

Using a Linear Model to Make a Prediction and an Estimate

Using a Linear Model to Make Estimates and Predictions

• 4 million visitors corresponds to v = 4• The corresponding v-coordinate is approx. t = 32•According to the linear model, there were 4 million visitors in the year 1960 + 32 = 1992

Example

Solution

Lehmann, Intermediate Algebra, 3ed

whether a linear function would model it well.

•Situation 1 Close to line-describes a linear function•Situation 2 & 3 Points do not lie close to one line•A linear model would not describe these situations

Section 2.1

Consider the scattergrams. Determine

Slide 9

Deciding Whether to Use a Linear Function to Model Data

When to Use a Linear Function to Model Data

Situation 1 Situation 2 Situation 3

Example

Solution

Lehmann, Intermediate Algebra, 3edSection 2.1

The wild Pacific Northwest salmon populations are listed in the table for various years.

1. Let P be the salmon

Slide 10

Intercepts of a Model; Model BreakdownIntercepts of a Model and Model Breakdown

population (in millions) at t years since 1950. Find a linear model that describes the situation.

•Data is described in terms of P and t in a table•Sketch a scattergram (see the next slide)

Example

Solution

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 11

Intercepts of a Model; Model BreakdownIntercepts of a Model and Model Breakdown

2. Find the P-intercept of the model. What does it mean?

3. Use the model to predict when the salmon will become extinct.

Example Continued

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 12

Intercepts of a Model; Model BreakdownIntercepts of a Model and Model Breakdown

• P- intercept is (0, 13)• When P = 13, t = 0 (the year 1950)• According to the model, there were 13 million

salmon in 1950• T-intercept is (45, 0)• When P = 0, t = 45 (the year 1950 + 45 = 1995• Salomon are still alive today• Our model is a false prediction

Solution

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 13

DefinitionIntercepts of a Model and Model Breakdown

For situations that can be modeled by a function whose independent variable is t:

when we part of the model whose t-coordinates are not between the t-coordinates of any two data points.

Definition

We perform interpolation

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 14

DefinitionIntercepts of a Model and Model Breakdown

We perform extrapolation when we use a part of the model whose t-coordinates are not between the t-coordinates of any two data points.

When a model gives a prediction that does not make sense or an estimate that is not a good approximation, we say that model breakdown has occurred.

Definition

Definition

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 15

Modifying a ModelIntercepts of a Model and Model Breakdown

In 2002, there were 3 million wild Pacific Northwest salmon. For each of the following scenarios that follow, use the data for 2002 and the data in the table to sketch a model. Let P be the wild Pacific Northwest salmon population (in millions) at t years since 1950.

1. The salmon population levels off at 10 million.

2. The salmon become extinct.

Example

Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 16

Modifying a ModelIntercepts of a Model and Model Breakdown

Solution