modeling with linear functions chapter 2. using lines to model data section 2.1
TRANSCRIPT
![Page 1: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/1.jpg)
Modeling with
Linear Functions
Chapter 2
![Page 2: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/2.jpg)
Using Lines to Model Data
Section 2.1
![Page 3: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/3.jpg)
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
![Page 4: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/4.jpg)
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
![Page 5: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/5.jpg)
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
![Page 6: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/6.jpg)
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
![Page 7: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/7.jpg)
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
![Page 8: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/8.jpg)
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
![Page 9: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/9.jpg)
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
![Page 10: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/10.jpg)
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
![Page 11: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/11.jpg)
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
![Page 12: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/12.jpg)
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
![Page 13: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/13.jpg)
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
![Page 14: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/14.jpg)
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
![Page 15: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/15.jpg)
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
![Page 16: Modeling with Linear Functions Chapter 2. Using Lines to Model Data Section 2.1](https://reader034.vdocument.in/reader034/viewer/2022051401/56649ee15503460f94bf239e/html5/thumbnails/16.jpg)
Lehmann, Intermediate Algebra, 3edSection 2.1 Slide 16
Modifying a ModelIntercepts of a Model and Model Breakdown
Solution