Purchasing a Used Car Algebra 2

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Abigail Casey
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1 Purchasing a Used Car Algebra 2 (This lesson was developed by Pam Hayes of J.M. Robinson High, Betty Anne Shearin of Columbia High, and Elizabeth Shepard of Hiedi Trask Senior High during the Leading to Success in Algebra 2 Workshop in.) Algebra 2 Goals Describe graphically, algebraically, and verbally real-world phenomena as functions; identify independent and dependent variables. (3.01) Translate between graphic, algebraic, and verbal representations of relations. (3.02) Write and graph eponential functions of the form f ( ) = a b. (3.15) Write and interpret an equation of a curve (eponential) which models a set of data. (4.01) Find the equation of the curve of best fit (eponential) for a set of data. Interpret the constants, coefficients, and bases in the contet of the data. Check the equation for goodness-of-fit and use the equation for predictions. (4.02) Use eponential equations of the form f ( ) = a (1 + r), where r is given as a rate of growth or decay to solve problems. (4.03) Materials 1. Copy of the handout for each student. 2. Graphing calculator for each student. 3. Access to the web site: (Kelly Blue Book web site). Activity One: Research on Car Prices 1. Discuss with students what will be the independent variable and what will be the dependent variable if discussing the value of a car and its age. 2. Students should use the Internet to find the web page: and follow the link to Private Party Value. (This active link is located in the body of tet with a heading of Used Car Values for Shoppers. ) Once into the link, students can select the type car, then the brand, and then the specific model. We want to use cars that have been in production for at least eight years. 3. Once each person has selected a car type, brand, and specific model and picked out the features desired, you will need to enter zip code and select condition. Use the mileage given by default (that will change with year). Always use the same condition and features. Create a data set by writing down each year (year car was made) and epected cost. 4. Several sample data sets are listed below. These can be used instead of the Internet activity. This data was collected in September 2003 from In this lesson plan, the Ford Taurus data will be used as the eample. Ford Taurus LX Sedan 4 door in good condition Year Value $ Purchasing a Used Car 1 Leading to Success in Algebra 2 Workshop

2 Toyota Camry LE Sedan 4 door in good condition Year Value $ Porsche Carrera Cabriolet 2 door in good condition Year Value $ Activity Two: Scatter Plot and Data Analysis 1. Students should first identify the independent and dependent variables in these tables. Just by using the web site for the data collection students can identify the value is produced as a function of the age of the car. 2. Using the calculator, put data into lists, year into L1 and Value into L2. 3. Since we want the data set to contain the ordered pairs (years old, value) not (year made, value) transform the eisting data into the desired form let years old=current year year made. This can be done in the table or through the lists in the calculator. List directions follow: Go to the top of L1 and type 2004-L1. (Subtract 2004 so the model year 2003 is thought to be a year old.) 4. Create a scatter plot of the data using age from L1 as the -value and value from L2 as the y-value. 5. The scatter plot verifies that the value of the car is decreasing over time. We need to consider which function best models the data. Students can get a feel for the best model by testing out the possible functions: linear, quadratic, or eponential. You can either Project the scatter plot against the chalkboard or whiteboard and sketch on the chalk board or whiteboard each of the possible functions over the scatter plot or Use Purchasing a Used Car (a power point file) to see slides of the data with linear, quadratic, and eponential functions superimposed over the data. Purchasing a Used Car 2 Leading to Success in Algebra 2 Workshop

3 The eponential will be the function of choice. Quadratic fits well for some of the data, but the function would fail to project reasonable values for cars older than Once the eponential function is selected as the model, the STAT Calc menu leads us to fit the data with an eponential regression model. The following screens document these steps. The eponential model is C ( ) = 11302(0.8603). 7. To determine if this model is a good model, investigate the residuals for each of the data points. A residual is defined to be the distance between a data point and the corresponding point on the model. The value of a residual is calculated by: ydata y Model for each -value of the data. The residual should measure the error in the model. To complete this process on the calculator, find values of each residual using the data in the lists. At the top of L3 input the formula L2-Y1 (L1). This takes the y-values of the data from L2 and subtracts the corresponding y-values from the function with Y1 (L1). Some residuals are positive which means the data point is above the curve. Some residuals are large which tells us that the data point is far from the model. To get an overall look at the information from the residuals, do a residual plot that plots the ordered pairs(, Data residual ). The residual plot on the calculator pairs L1 with L3. In this data set, the residual plot is: This residual plot has some pattern in the points usually we look for a residual plot to look random without pattern. When other types of models are used linear and quadratic function, the residual plot shows a more distinct pattern. The eponential Purchasing a Used Car 3 Leading to Success in Algebra 2 Workshop

4 model has the best result. Also notice, in this plot we see the greatest error in the cars most recently made and the least error in the oldest cars. 8. The model for this data is C ( ) = 11302(0.8603). What information is in this model? a. If = 0 then we know the value (what we could sell it for) is $11302 for a brand new Ford Taurus LX Sedan 4 door in good condition. b. The base of the eponent is This base tells us that as we move from one year s model to the model of the previous year the value is multiplied by The value is 86.03% of the value of a car one year newer. c. Usually with rates we turn that around and think about multiplying by 1 r which in this case would be = or the value drops by 13.97% with each year that goes by. d. Using this model we can predict the value of a 20-year-old Ford Taurus LX 20 Sedan 4 door in good condition. In this case, $11302(0.8603) = $ Conceivably, you could find the value of a 50-year-old car, but that would not be reasonable since this car was not made 50 years ago. Also, after 25 years, cars become antiques and the value should not be consistent with the model. 9. To determine the year that the car would be $100, we either need to use the intersect option on the calculator or we need to use logarithms. Both methods will be discussed. a. Using the intersection option on the calculator put your model in Y1 and the equation y = 100 in Y2. Adjust your window to show -values from 0 to 35 and a much smaller range of y-values the point of intersection must show on the screen. We find the intersection occurs at (31.4,100) or a 31.4-year-old car (made in ) is worth $100. Again, this is outside our reasonable domain. We might answer that every Taurus is worth something! b. Our answer should be similar if logarithms are used. The method follows. Purchasing a Used Car 4 Leading to Success in Algebra 2 Workshop

5 100 = 11302(0.8603) 100 = (0.8603) log = log(0.8603) log = log(0.8603) log = log(0.8603) 31.4 = Purchasing a Used Car 5 Leading to Success in Algebra 2 Workshop

6 Student Handout Purchasing a Used Vehicle Access the web site: and click on the hyper link Private Party as shown to the left. USED CAR VALUES FOR SHOPPERS Shopping for a used car? Check out Blue Book values before you buy. Choose Used Car Retail Value if you intend to buy from a dealer. Choose Private Party Value if you intend to buy from another consumer. Chose a vehicle in which you are interested. To be consistent, determine the features you desire and use the same features for each year s model, and use the default mileage given. Complete the table. Vehicle: Condition: Years Old Year Made Value 1. Determine which variable is independent and which is dependent. Enter the data into lists on your calculator. Use Years Old and Value as variables. 2. Create a scatter plot of the data and determine which function (linear, quadratic, or eponential) best models the data. 3. Find the regression function. Identify what and y represent. 4. Discuss the goodness of fit of this model. 5. What information does this model give us about the values of cars over time? What is the specific meaning of each constant in the model? What is a reasonable domain for the model? (Write in complete sentences.) 6. What is the rate of growth or decay in the value of the vehicle? 7. Predict the value of a 20-year-old vehicle. 8. What year will the vehicle be nearly worthless (less than $100)? Purchasing a Used Car 6 Leading to Success in Algebra 2 Workshop

Section 1.1: Functions and Models

Section 1.1: Functions and Models Definition: A function is a rule that assigns to each element of one set (called the domain) exactly one element of a second set (called the range). A function can be