The table shows the speed of a vehicle in miles per hour, x, and the fuel economy of the vehicle in miles per gallon, f(x).
Fuel Economy of Vehicle
Speed, x (miles per hour)
Fuel Economy, f(x) (miles per gallon)
45 |32.2
50 |33.9
55 |35.0
60 |33.9
65 |32.2
70 |29.1
75 |27.0
Which quadratic function best models the data in the table?
f(x) = -0.02x + 2.15x - 24.77
f(x) = -x + 3.3x - 1,992.8
f(x) = -0.03x7 + 3.3x - 55.75
f(x) = -x + 110x - 2,990
To determine the quadratic function that best models the data in the table, we need to find the equation in the form f(x) = ax^2 + bx + c that represents the relationship between the speed x and the fuel economy f(x).
We can start by examining the given data points in the table. Notice that the fuel economy varies across different speeds. This means that the equation should have a quadratic term (x^2) to capture the curvature in the relationship.
To find the quadratic function, we will use the method of least squares. We need to minimize the sum of the squared differences between the actual fuel economy values and the predicted fuel economy values based on the proposed quadratic equation.
Let's calculate the squared differences for each data point using the equation options given:
1) For f(x) = -0.02x + 2.15x - 24.77:
45: (32.2 - (-0.02(45)^2 + 2.15(45) - 24.77))^2 = (32.2 - 64.27)^2 = 129.96
50: (33.9 - (-0.02(50)^2 + 2.15(50) - 24.77))^2 = (33.9 - 64.23)^2 = 102.49
55: (35.0 - (-0.02(55)^2 + 2.15(55) - 24.77))^2 = (35.0 - 68.91)^2 = 130.42
60: (33.9 - (-0.02(60)^2 + 2.15(60) - 24.77))^2 = (33.9 - 74.73)^2 = 179.94
65: (32.2 - (-0.02(65)^2 + 2.15(65) - 24.77))^2 = (32.2 - 80.59)^2 = 228.28
70: (29.1 - (-0.02(70)^2 + 2.15(70) - 24.77))^2 = (29.1 - 86.49)^2 = 281.64
75: (27.0 - (-0.02(75)^2 + 2.15(75) - 24.77))^2 = (27.0 - 92.43)^2 = 339.38
Let's repeat the same calculations for the other equation options.
2) For f(x) = -x + 3.3x - 1,992.8:
45: (32.2 - (-(45) + 3.3(45) - 1,992.8))^2 = (32.2 - 26.7)^2 = 30.25
50: (33.9 - (-(50) + 3.3(50) - 1,992.8))^2 = (33.9 - 28.5)^2 = 28.09
55: (35.0 - (-(55) + 3.3(55) - 1,992.8))^2 = (35.0 - 30.5)^2 = 20.25
60: (33.9 - (-(60) + 3.3(60) - 1,992.8))^2 = (33.9 - 32.4)^2 = 7.84
65: (32.2 - (-(65) + 3.3(65) - 1,992.8))^2 = (32.2 - 34.3)^2 = 4.41
70: (29.1 - (-(70) + 3.3(70) - 1,992.8))^2 = (29.1 - 36.2)^2 = 11.56
75: (27.0 - (-(75) + 3.3(75) - 1,992.8))^2 = (27.0 - 38.1)^2 = 11.56
3) For f(x) = -0.03x^7 + 3.3x - 55.75:
45: (32.2 - (-0.03(45)^7 + 3.3(45) - 55.75))^2 = (32.2 - 440.39)^2 = 370,205.64
50: (33.9 - (-0.03(50)^7 + 3.3(50) - 55.75))^2 = (33.9 - 625.65)^2 = 369,366.56
55: (35.0 - (-0.03(55)^7 + 3.3(55) - 55.75))^2 = (35.0 - 892.69)^2 = 369,313.36
60: (33.9 - (-0.03(60)^7 + 3.3(60) - 55.75))^2 = (33.9 - 1,250.69)^2 = 369,262.84
65: (32.2 - (-0.03(65)^7 + 3.3(65) - 55.75))^2 = (32.2 - 1,708.4)^2 = 369,119.24
70: (29.1 - (-0.03(70)^7 + 3.3(70) - 55.75))^2 = (29.1 - 2,270.71)^2 = 368,658.92
75: (27.0 - (-0.03(75)^7 + 3.3(75) - 55.75))^2 = (27.0 - 2,934.25)^2 = 368,551.56
4) For f(x) = -x + 110x - 2,990:
45: (32.2 - (-(45) + 110(45) - 2,990))^2 = (32.2 - 4,195)^2 = 14,731,329.64
50: (33.9 - (-(50) + 110(50) - 2,990))^2 = (33.9 - 5,410)^2 = 16,769,321
55: (35.0 - (-(55) + 110(55) - 2,990))^2 = (35.0 - 6,465)^2 = 21,972,030.25
60: (33.9 - (-(60) + 110(60) - 2,990))^2 = (33.9 - 7,360)^2 = 27,995,304.4
65: (32.2 - (-(65) + 110(65) - 2,990))^2 = (32.2 - 8,095)^2 = 32,782,240.25
70: (29.1 - (-(70) + 110(70) - 2,990))^2 = (29.1 - 8,670)^2 = 36,257,355.24
75: (27.0 - (-(75) + 110(75) - 2,990))^2 = (27.0 - 9,085)^2 = 38,684,729
Based on the squared differences calculated for each equation option, we can conclude that the quadratic function that best models the data in the table is:
f(x) = -x + 3.3x - 1,992.8