2.2: Linear Models

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INTRODUCTION

In Collaboration M.1, you were introduced to the idea of a mathematical model. There are many different types of mathematical models. We will first look at a type called a linear model, which is both one of the simplest and most important kinds of models. Any situation that has a “constant rate of change” is a linear model. There are many common examples of linear models, including sales tax, linear depreciation, hourly salary, and the grade (steepness) of a road.

In Preparation M.2, various definitions for linear models were presented. In this collaboration, we will use these definitions, learning about them in more detail. Check-in with your group members to make sure everyone understands the following definitions:

Definition: The slope of a line is the ratio of the change in the vertical value (sometimes called output, y-value, dependent variable, or response variable) to the change in the horizontal value (sometimes called input, x-value, independent variable, or explanatory variable) as we move between any two points on the line. The slope of a line is therefore a rate of change. Often it is referred to as the ratio of the rise of the graph to the run of the graph, or “rise over run.” The slope can also be described as the rate at which the output values are changing (increasing or decreasing) as the input value increases by 1. It can be found by using any two points on the line. (Note: By convention, the input is on the horizontal axis and the output on the vertical axis.)

Definition: The vertical intercept of a line is the point at which the line intersects the vertical axis, where the input value is zero. This is also called the initial value if the input starts out at a value of zero.

Definition: The horizontal intercept of a line is the point at which the line intersects the horizontal axis, where the output is zero.

Definition: A linear model has a constant rate of change (slope). (This is not true of all models and their corresponding equations.)

SPECIFIC OBJECTIVES

By the end of this collaboration, you should understand that

the slope of a linear model is a rate of change.
a linear model has a constant rate of change.
the vertical intercept of a linear model is an initial value.
linear models can be used to approximate real-life situations, though they are often not exact.
data that changes in an approximately linear manner can be modeled by a linear regression model (trendline).

By the end of this collaboration, you should be able to

create a table, graph and/or equation from a verbal description of a linear model.
move freely between the four different modes of looking at linear models (words, tables, graphs and equations).
find and interpret the slope of a linear model from the four different representations.
find and interpret the horizontal and vertical intercepts of a linear model from the four different representations.
use a spreadsheet to create a scatterplot and linear regression model (trendline) from data.

PROBLEM SITUATION 1: ENERGY BILLS

The company that provides electricity to people in and around New Orleans, Louisiana, is called Entergy. The table below shows an estimate of monthly rates charged by Entergy.¹² Use this table to respond to the questions below.

Summer Billing Cycles

(May – October)

Winter Billing Cycles

(November – April)

Energy Charges:

6.002¢ per kWh
Monthly Customer Charge: $8.07

Energy Charges:

First 800 kWh per month at 6.002¢ per kWh
Over 800 kWh per month at 4.766¢ per kWh
Monthly Customer Charge: $8.07

For this collaboration, we will only be using the summer billing cycle. Notice that the monthly service charge of $8.07 is applied to each monthly bill, no matter how much electricity is used.

(1) Suppose a customer uses no electricity in July. What would her monthly bill be?

(2) A customer uses 900 kWh in July. What would her monthly bill be?

(3) Another customer uses 901 kWh in July. What would her monthly bill be?

(4) What is the difference between outputs in your answers to Question 2 and Question 3? What is the difference between the inputs you used to answer Questions 2 and 3? Now use these differences to calculate the slope.

(5) Explain why a linear model would represent Entergy’s summer billing charges for any customer.

(6) Considering how you calculated your answers to Questions 2 and 3, find a linear equation that models any Entergy customer’s bill. Use k for energy usage (in kWh), and B for Entergy's charge (in dollars).

(7) Sketch a graph of your linear model. Note: If completing this problem online, follow the instructions given online to create your graph.

(8) What is the vertical intercept of this model? How is it related to billing? Write your answer as a point (k, B), and then write a sentence for your explanation.

(9) Does this model have a horizontal intercept? What does it mean?

(10) Suppose a family makes a quarterly (3-month) budget to plan for upcoming expenses. This family estimates they will use an average of 1000 kWh per month in July, August and September. How much money must the family set aside in the budget so that they will be able to pay for the electricity they will use during the three-month period? Round to the nearest dollar, since budgets are only estimates.

PROBLEM SITUATION 2: CELL PHONE DEPRECIATION

Smartphones, like many other manufactured goods, depreciate on a daily basis. Depreciation means how much an item (e.g. a smartphone) loses in value compared to the price that was originally paid for it from the manufacturer. There tends to be two main periods in a smartphone's life when it sees the biggest drops in value: 1. the period of time after the phone is initially purchased, and 2. when the manufacturer launches a new model. The decrease in value of an item over time is called depreciation.

In September 2019, the iPhone 11 256GB was released with a price of $849.¹³ Twelve months after the launch, it was found that the value of iPhone 11s depreciated about 37% (which is actually better than most smartphone depreciation!).¹⁴

Suppose you bought a new iPhone 11 256GB in September 2019 for $849. The phone would soon start depreciating in value. The following table shows the approximate value of your iPhone 11, by month from September 2019.

Month

September 2019

October

2019

November 2019

December 2019

January 2020

February 2020

Phone Value in Dollars

849

800

780

750

720

699

(11) It can be easier to work with models when the data are numerical. Modify the first row of the table to change the months into numbers. Compare your answers with the rest of your group and the class.


Phone Value in Dollars	849	800	780	750	720	699

(12) Now, use the table from Question 11 to create a graph of the data. Remember to label the axes. Note: If completing this problem online, follow the instructions given online to create your graph.

(13) Does the data appear to be linear? Explain your answer.

(14) What is the vertical intercept of the graph? What does this mean in relation to the situation?

(15) Answer the next three questions using the assumption that the data is approximately linear.

(a) Approximate the slope of the linear model. (Hint: When data are only approximately linear, be careful when choosing the two points you use to calculate the slope. Often a good strategy is to use the first point and last point since they represent the total change in the data.)

(b) What is the meaning of this slope in this situation?

(16) Write an equation that gives a linear model for the value of the cell phone from Sept 2019 through February 2020. Use V for the value in dollars, and m for the number of months after Sept 2019, and use the slope from Question 15c.

(17) Suppose you want to sell your phone before it is worth less than $300. Using your model, when is the last month you could sell it and get at least $300?

(18) What is the horizontal intercept of your model? What is the meaning of this in the context of this problem? Is this reasonable?

PROBLEM SITUATION 3: CELL PHONE DEPRECIATION USING TRENDLINES

We just saw that a linear model can be used to model data that is reasonably, but not perfectly, linear. In Problem Situation 2, we used a rough linear model that may not have represented the data most accurately. There are ways to create a linear model that represents the data more carefully. Your instructor will show you how this is done now in a spreadsheet application, such as Microsoft Excel or Google Sheets. We will be using the table created in Question 11:

Months after Sept 2019	0	1	2	3	4	5
Phone Value in Dollars	849	800	780	750	720	699

(19) Create a scatterplot of the data using a spreadsheet application. Does the data look linear?

(20) We will now use the spreadsheet application to calculate the trendline. The trendline is a linear model representing the trend of the data. When using technology, the trendline is often called the line of best fit, which is the line that comes closer to all the data points, on average, than any other possible line, for data that isn’t perfectly linear. If possible, display the trendline along with the scatterplot.

MAKING CONNECTIONS

Record the important mathematical ideas from the discussion.

In this collaboration, we have looked at linear data in four different ways. Each way can be useful in different situations.

We have seen VERBAL DESCRIPTIONS, such as “Energy Charges 6.002¢ per kWh [and a] Monthly Customer Charge [of] $8.07.” Many real-life problems start out as situations that we can describe in words. But we can also try to use alternate representations that make the data more clear, and easier to work with mathematically.

We have also seen NUMERICAL REPRESENTATIONS of data, which means the data is presented in a table that allows us to organize and carefully look over the data, such as the following:

Months after Sept 2019	0	1	2	3	4	5
Phone Value in Dollars	849	800	780	750	720	699

We have also seen GRAPHICAL REPRESENTATIONS of data, such as the graph given in Question 2 in Preparation M.2:

Graph showing an example of the Linear Model.

Y-axis ranges from negative 10 to 15.
X-axis ranges from 0 to 40.

0 = negative 10
20 = 0
40 = 10

Notice that when data is presented graphically, it is very easy to see if it appears to be linear, and to read information such as intercepts from the graph.

Finally, we have created ALGEBRAIC MODELS of the data, which usually means to give an equation representing the data, such as V = 849 − 30m. Once we have an algebraic model, we can use it to answer questions about the data using the given information and solving for unknown quantities - for example, find the number of months it takes for a $849 cellphone to depreciate to $0 knowing that it depreciates at the rate of $30 per month.

___________________________________

¹² https://www.entergy-louisiana.com/your_home/price/

¹³ https://www.macrumors.com/2021/07/16/iphone-12-depreciates-less-than-iphone-11/

¹⁴ https://www.decluttr.com/phone-depreciation/

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