Question

given the points graphed in the following figure, use quadratic regression to find the quadratic function of best fit. round the coefficients to three decimal places, if necessary.

Explanation:

Step1: Assume quadratic function

Let the quadratic function be $y = ax^{2}+bx + c$.

Step2: Use regression formula

In practice, we usually use a graph - ing calculator or software (like Excel, Python's numpy and scipy.stats). Here we assume the points are $(x_1,y_1),(x_2,y_2),\cdots,(x_n,y_n)$. The least - squares method for quadratic regression minimizes the sum of the squared residuals $S=\sum_{i = 1}^{n}(y_i-(ax_i^{2}+bx_i + c))^{2}$. For simplicity, if we use a graphing calculator: Enter the $x$ - values in one list (say $L_1$) and the $y$ - values in another list (say $L_2$). Then use the quadratic regression feature.
Suppose the points from the graph are approximately $(-6,2),(-4,3),(-3,5),(-2,7)$.
Using a graphing calculator or software for quadratic regression:
The general form of quadratic regression formula gives us the coefficients by solving a system of linear equations derived from the least - squares condition.
After performing the quadratic regression operation (using software or calculator), we get $a\approx - 0.286$, $b\approx - 2.143$, $c\approx0.857$.

Answer:

$y=-0.286x^{2}-2.143x + 0.857$