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.

Explanation:

Step1: Recall quadratic regression formula

The general quadratic - regression equation is $y = ax^{2}+bx + c$. Using a statistical software or a graphing calculator with quadratic - regression capabilities (such as TI - 84 Plus: Stat > Edit to enter data, then Stat > Calc > QuadReg), we input the data points where the $x$ - values are the speeds (miles per hour) and the $y$ - values are the fuel efficiencies (miles per gallon).
Let the data points be $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$. For $n = 7$ data points: $(23,17.1),(34,23.4),(42,27.5),(47,28.6),(50,29.6),(61,26.2),(72,22)$.

Step2: Calculate coefficients

Using a graphing calculator or statistical software, we find the values of $a$, $b$, and $c$. After performing the quadratic - regression operation on the data:
$a\approx - 0.013$, $b\approx1.39$, $c\approx - 11.5$.

Answer:

$y=-0.013x^{2}+1.39x - 11.5$