1. Chapter 2
Modeling with
Linear Functions

2. Using Lines to Model Data
Section 2.1
Using Lines to Model Data

3. Using Lines to Model Data
Scattergrams
Example
The number of Grand Canyon visitors is listed in the table for various years. Describe the data.
Solution
Let v be the number (in millions) of visitors
Let t be the number of years since 1960
Section 2.1
Slide 3

4. Sketch a line that comes close to (or on) the data points.
Using Lines to Model Data
Scattergrams
Example Continued
Sketch a line that comes close to (or on) the data points.
The graph on the left does the best job of this.
Section 2.1
Slide 4

5. Definitions Definition
Linear Models
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.
Definition
Section 2.1
Slide 5

6. Definitions Definition Property
Linear Models
Definition
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
Property
Section 2.1
Slide 6

7. Use a linear model to predict the number of visitors in 2010.
Using a Linear Model to Make a Prediction and an Estimate
Using a Linear Model to Make Estimates and Predictions
Example
Use a linear model to predict the number of visitors in 2010.
Solution
Year 2010 corresponds to t = 50: – 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
Section 2.1
Slide 7

8. Use a linear model to estimate the year there ware 4 million visitors.
Using a Linear Model to Make a Prediction and an Estimate
Using a Linear Model to Make Estimates and Predictions
Example
Use a linear model to estimate the year there ware 4 million visitors.
Solution
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 = 1992
Section 2.1
Slide 8

9. Consider the scattergrams. Determine
Deciding Whether to Use a Linear Function to Model Data
When to Use a Linear Function to Model Data
Example
Consider the scattergrams. Determine
Situation Situation Situation 3
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
Solution
Section 2.1
Slide 9

10. Intercepts of a Model; Model Breakdown
Intercepts of a Model and Model Breakdown
Example
The wild Pacific Northwest salmon populations are listed in the table for various years.
1. Let P be the salmon
population (in millions) at t years since 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)
Solution
Section 2.1
Slide 10

11. Intercepts of a Model; Model Breakdown
Intercepts of a Model and Model Breakdown
Example Continued
2. Find the P- intercept of the model. What does it mean?
3. Use the model to predict when the salmon will become extinct.
Section 2.1
Slide 11

12. Intercepts of a Model; Model Breakdown
Intercepts of a Model and Model Breakdown
Solution
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 = 1995
Salomon are still alive today
Our model is a false prediction
Section 2.1
Slide 12

13. Intercepts of a Model and Model Breakdown
Definition
Intercepts of a Model and Model Breakdown
Definition
For situations that can be modeled by a function whose independent variable is t:
We perform interpolation
when we part of the model whose t-coordinates are not between the t-coordinates of any two data points.
Section 2.1
Slide 13

14. Intercepts of a Model and Model Breakdown
Definition
Intercepts of a Model and Model Breakdown
Definition
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.
Definition
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.
Section 2.1
Slide 14

15. Intercepts of a Model and Model Breakdown
Modifying a Model
Intercepts of a Model and Model Breakdown
Example
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.
Section 2.1
Slide 15

16. Intercepts of a Model and Model Breakdown
Modifying a Model
Intercepts of a Model and Model Breakdown
Solution
Section 2.1
Slide 16