Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.1, Slide 1 Chapter 2 Modeling with Linear Functions

Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.1, Slide 1

Chapter 2Modeling with Linear Functions


2.1 Using Lines to Model Data


Example: Using a Graph to Describe an Authentic Situation

The Grand Canyon is a beautiful landmark, yet the difficulty of finding a parking spot can detract from visitors’ enjoyment. The numbers of Grand Canyonvisitors are listed in the table for various years. Describe the data with a graph.


Solution

Let v be the number (in millions) of visitors in the year that is t years since 1960. We can describe the data with a table of values for v and t.


Solution

Plot the (t, v) data points shown in the table. Let the horizontal axis be the t-axis and the vertical axis be the v-axis.


Scattergram

A graph of plotted ordered pairs is called a scattergram. A scattergram should have scaling on both axes and labels indicating the variable names and scale units.


Approximately Linearly Related

We can sketch a line that comes close to (or on) the data points of the graph from the previous example. Since the points in the scattergram lie close to (or on) a line, then we say that the relevant variables are approximately linearly related.



One possible line that comes close to (or on) the data points.



A few of the many lines that come close to (or on) the data points.


Model

Definition

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


Linear model

Definition

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


Example: Using a Linear Model to Make a Prediction and an Estimate

1. Use the linear model shown on the next slide to predict the number of visitors in 2018.

2. Use the linear model to estimate in what year there were 4 million visitors.


Solution

1. 2018 – 1960 = 58, so t = 58. To estimate the number of visitors, locate the point on the linear model where the t-coordinate is 58. The corresponding v-coordinate is about 5.5, sothere will be 5.5 million visitors in 2018.


Solution

2. To find the year when there were 4 million visitors, locate the point on the linear model where the v-coordinate is 4. The coordinating t-coordinate is about 37. So, according to the linear model, there were 4 million visitors in 1960 + 37 = 1997.


Using a Linear Function to Model Data

Warning

We create a scattergram of data to determine whether the relevant variables are approximately linearly related. If they are, we draw a line that comes close to (or on) the data points and use the line to make estimates and predictions. It is a common error to try to find a line that contains the greatest number of data points. Our goal is to find a line that comes close to all of the data points.


Example: Intercepts of a Model;

Model Breakdown

The percentages of cell phone users who send or receive text messages multiple times per day are shown in the table for various age groups.


Example: Intercepts of a Model;

Model Breakdown

1. Let p be the percentage of cell phone users at age a who send or receive text messages multiple times per day. Find a linear model that describes the relationship between a and p.

2. Find the p-intercept. What does it mean in this situation?

3. Find the a-intercept. What does it mean in this situation?


Solution

1. View the positions of the points in the scattergram. a and p appear approximately linearly related, so we sketch a line that comes close to the data points.


Solution

2. The p-intercept is (0, 106), or p = 106, when a = 0.

According to the model, 106% of newborns who use cell phones send or receive text messages multiple times per day.

Our model gives a false estimate for two reasons: Percentages cannot be larger than 100% in this situation, and newborns cannot send or receive text messages.


Solution

3. The a-intercept is (71, 0), or p = 0, when a = 71.

According to the model, no 71-year-old cell phone users send or receive text messages multiple times per day. This is a false statement.

A little research would show some 71-year-old cell phone users send or receive text messages multiple times per day.


Interpolation, extrapolation

Definition

For a situation that can be modeled by a function whose independent variable is t,

•We perform interpolation when we use a part of the model whose t-coordinates are between the t-coordinates of two data points.

•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.


Interpolation, extrapolation


Model Breakdown

Definition

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


Example: Modifying a Model

Additional research yields the data shown in the first and last rows of the table. Use the data and the following assumptions to modify the model found in the previous example.

•Children 3 years old and younger do not send or receive text message multiple times per day.•The percentage of cell phone users who send or receive text messages levels off at 5% for users over 80 years of age.•The age of the oldest cell phone user is 116 years.


Example: Modifying a Model


Solution

Recall that p is the percentage of cell phone users at age a years who send or receive multiple text messages per day. Taking into account the three assumptions, we sketch a scattergram and draw a model that comes close to the data points.

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Copyright © 2015, 2008, 2011 Pearson Education, Inc. Section 2.1, Slide 1 Chapter 2 Modeling with Linear Functions