Question

find the quadratic regression equation for the relationship of the horizontal distance and the height of the ball in the provided table. -6x² + 1.06x + 79 -18x² + 2.11x + 4.21 -0.12x² + 2.11x + 4.22 -0.06x² + 1.06x + 7.9

Explanation:

Step1: Recall the quadratic regression formula

The general form of a quadratic regression equation is \(y = ax^{2}+bx + c\). We can use a calculator or software (like Excel, TI - 84 Plus) with quadratic regression functionality. Input the \(x\) - values (distance: \(0,2,6,9,12,13,15\)) and \(y\) - values (height: \(4,8.4,12.1,14.2,13.2,10.5,9.8\)).

Step2: Analyze the coefficients

When we perform quadratic regression on the given data:

• For the coefficient \(a\):
• If \(a=-6\) (as in \(y=-6x^{2}+1.06x + 79\)), when \(x = 0\), \(y=79\) which is not consistent with the table (\(y(0)=4\)).
• If \(a=-18\) (as in \(y=-18x^{2}+2.11x + 4.21\)), the function will decrease too rapidly.
• For the constant term \(c\):
• When \(x = 0\), \(y=c\). From the table \(y(0) = 4\).
• In the equation \(y=-0.06x^{2}+1.06x + 7.9\), when \(x = 0\), \(y = 7.9

eq4\)

• In the equation \(y=-0.12x^{2}+2.11x + 4.22\), when \(x = 0\), \(y = 4.22\approx4\) (considering rounding in regression calculations).

Answer:

\(-0.12x^{2}+2.11x + 4.22\)