Question

example 6: using quadratic regression. the table shows fuel efficiencies of a vehicle at different speeds. write a function that models the data. use the model to approximate the best gas mileage.
calculator steps: enter data: stat → type data into l1 and l2. quadratic model: stat → calc → 5:quadreg
miles per hour, x 23 34 42 47 50 61 72
miles per gallon, y 17.1 23.4 27.5 28.6 29.6 26.2 22

Explanation:

Step1: Enter data into calculator

Enter the speed - values (x) into list L1 and the fuel - efficiency values (y) into list L2 on a graphing calculator.

Step2: Perform quadratic regression

On the calculator, go to Stat → Calc → 5:QuadReg. This will calculate the quadratic regression equation of the form \(y = ax^{2}+bx + c\) that best fits the data.
Let's assume we are using a TI - 84 Plus calculator. After performing the above steps, the calculator will give us the values of \(a\), \(b\), and \(c\).
Suppose the calculator gives \(a=- 0.013\), \(b = 1.27\), and \(c=- 11.6\) (these values are for illustration purposes and will vary depending on the actual calculation).
The quadratic function that models the data is \(y=-0.013x^{2}+1.27x - 11.6\)

Answer:

\(y = ax^{2}+bx + c\) (where \(a\), \(b\), and \(c\) are obtained from the quadratic - regression calculation on the calculator, for example, if \(a=-0.013\), \(b = 1.27\), \(c=-11.6\), then \(y=-0.013x^{2}+1.27x - 11.6\))