UNIT 6 DESCRIBING DATA Lesson 2: Working with Two Variables. Instruction. Guided Practice Example 1
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1 Guided Practice Eample 1 Andrew wants to estimate his gas mileage, or miles traveled per gallon of gas used. He records the number of gallons of gas he purchased and the total miles he traveled with that gas. Gallons Miles Create a scatter plot showing the relationship between gallons of gas and miles driven. Which is a better estimate for the function that relates gallons to miles: = 1 or = 22? How is the equation related to his gas mileage? U6-11 SWB p. 69
2 1. Plot each point on the coordinate plane. Let the -ais represent the number of gallons and the -ais represent the number of miles. Miles Gallons U6-111
3 2. Graph the first function on the coordinate plane. The function = 1 is linear, so onl two points are needed to draw the line. Evaluate the equation at two values of, such as and 1, and draw a line through these points on the scatter plot. = 1 First equation = 1() = Substitute for. = 1(1) = Substitute 1 for. Two points on the line are (, ) and (1, ). Miles = Gallons U6-112
4 3. Graph the second function on the same coordinate plane. The function = 22 is also linear, so onl two points are needed to draw the line. Evaluate the equation at two values of, such as and 1, and draw a line through these points on the scatter plot. = 22 Second equation = 22() = Substitute for. = 22(1) = 22 Substitute 1 for. Two points on the line are (, ) and (1, 22). Miles = 22 = Gallons U6-113
5 4. Identif which function comes closer to the data values. The function whose line comes closer to the data values is the better estimate for the data. The graph of the function = 22 goes through the points in the scatter plot, with some points falling above this line and some points falling below it. The function = 1 is not steep enough to match the data values. Therefore, the function = 22 is a better estimate of the data. 5. Interpret the equation of the function in the contet of the problem, using the units of the - and -aes. For a linear equation in the form = m + b, the slope (m) of the equation is the rate of change, or the change in over the change in. The -intercept (b) of the equation is the initial value. change in miles In this eample, is miles and is gallons. The slope is change in gallons. For the equation = 22, the slope of 22 is equal to 22 miles 1gallon. The gas mileage of Andrew s car is the miles driven per gallon of gas used. The gas mileage is equal to the slope of the line that fits the data. Andrew s car has a gas mileage of approimatel 22 miles per gallon. U6-114
6 Eample 2 The principal at Park High School records the total number of students each ear. The following table shows the number of students for each of the last 8 ears. Year Number of students Create a scatter plot showing the relationship between the ear and the total number of students. Show that the function = 6(1.5) is a good estimate for the relationship between the ear and the population. Approimatel how man students will attend the high school in ear 9? 1. Plot each point on the coordinate plane. Let the -ais represent the ear number and the -ais represent the number of students. Number of students 1, Year U6-115
7 2. Graph the given function on the coordinate plane. The function is = 6(1.5). Calculate the value of for at least five different values of. Start with =. Calculate the value of the function for at least four more -values that are in the data table. 6(1.5) = 6 1 6(1.5) 1 = (1.5) 3 = (1.5) 5 = (1.5) 7 = Plot these points on the same coordinate plane. Connect the points with a curve. Note: The new points are plotted in white. Number of students 1, Year U6-116
8 3. Compare the graph of the function to the scatter plot of the data. The graph of the function appears to be ver close to the points in the scatter plot. Therefore, = 6(1.5) is a good estimate of the data. 4. Use the equation to answer the question. Use the equation to estimate the population in ear 9. Evaluate the equation = 6(1.5) for ear 9, when = 9. = 6(1.5) 9 = The equation = 6(1.5) is a good estimate of the population. There will be approimatel 931 students in the school in ear 9. Eample 3 Thomas wants to know how long it will take his truck to stop based on how fast it is moving. The following table shows the distance the truck traveled after the brakes were applied while traveling at different speeds. Speed (mph) Stopping distance (ft) Create a scatter plot showing the relationship between the speed of the truck and the distance it takes to stop once the brakes are applied. Show that the function =.45 2 is a good estimate for the relationship between the speed and the stopping distance. About how far would it take the truck to stop if it were traveling at 7 mph? U6-117
9 1. Plot each point on the coordinate plane. Let the -ais represent the speed of the truck, and let the -ais represent the stopping distance Stopping distance (feet) Speed (mph) U6-118
10 2. Graph the given function on the coordinate plane. The function is = Calculate the value of for at least five different values of. Start with =, then calculate the value of the function for at least four more -values in the data table..45() 2 = 2.45(2) 2 = (3) 2 = (4) 2 = (6) 2 = 162 Plot these points on the same coordinate plane, then connect the points with a curve. Note: The new points are plotted as hollow dots Stopping distance (feet) Speed (mph) U6-119
11 3. Compare the graph of the function to the scatter plot of the data. The graph of the function appears to be ver close to the points in the scatter plot. Therefore, =.45 2 is a good estimate of the data. 4. Use the equation to answer the question. Use the equation to estimate the stopping distance when the truck is travelling 7 mph. Evaluate the equation =.45 2 for the speed of 7 mph, when = 7. =.45(7) 2 = 22.5 The equation =.45 2 is a good estimate of the stopping distance of the truck. The stopping distance is approimatel 22.5 feet when the truck is traveling 7 mph. U6-12
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