The table below shows the height of a ball x seconds after being kicked.

what values, rounded to the nearest whole number, complete the quadratic regression equation that models the data?
f(x) = x^2 + x + 0
based on the regression equation and rounded to the nearest whole number, what is the estimated height after 0.25 seconds?

feet

Part A:

The general form of a quadratic function is given by:



Given that the table shows the height (y) of a ball x seconds after being kicked.
Thus, the time represents the x-values while the height represents the y-values.

From the table it can be seen that when the time is 0, the height is also 0, thus:



Also from the table it can be seen that when the time is 1, the height is 65, thus:



Also from the table it can be seen that when the time is 2, the height is 95, thus:



Multiplying equation (2) by 2, we have:
2a + 2b = 130 . . . (4)

Subtracting equation (4) from equation (3) gives:
2a = -35
or a = -35 / 2 = -17.5

Substituting for a into equation (2) gives:
-17.5 + b = 65
or b = 65 + 17.5 = 82.5

Therefore, the quadratic regression equation that models the data is given by f(x) = -18x^2 +83x + 0

Part B:

Using the function obtained in part a, with x = 0.25, we have



Therefore, the estimated height after 0.25 seconds is 20 feet.

A quadratic regression equation is of the form
y = c₃x² + c₂x +c₁

Coefficients c₁,c₂,c₃ are determined by minimizing the least squared error

This means that


This creates the matrix equation for determining the c-coefficients:


For the given problem,
the matrix equation is


The solution for the coefficients is
c₁ = -0.3571
c₂ = 81.0714
c₃ = -16.4286

The graph of the fitted data is shown below.

When x = 0.25,
y = c₃(0.25)² + c₂(0.25) + c₁ = 18.884

The regression equation is
f(x) = -16.4286x² + 81.0714x - 0.3571
When rounded to the nearest whole number,
f(x) = -16x² + 81x + 0
and
f(0.25) = 19.25

Answers:
f(x) = -16x² + 81x + 0  (approximated)
f(0.25) = 19.25  (approximated)

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