Question

3. Patrick kicks a soccer ball from the ground into the air. When the ball is a horizontal distance of 12 feet from

him, it is 14 feet above the ground. When the ball is 32 feet from him, it is 7 feet above the ground.
Use quadratic regression on a graphing calculator to write a function for the height of the ball in terms of
its horizontal distance Round to the nearest thousandth (

Answer 1

The function for the height of the ball in terms of its horizontal distance is

`y = -(91)/(1920)\cdot x^(2)+(833)/(480)\cdot x`

Explanation:

From Physics, we know that ball describes a parabolic motion, in which trajectory is described by the following second-order polynomial (quadratic function):

`y = a\cdot x^(2)+b\cdot x + c` (Eq. 1)

Where:

`x` - Horizontal distance, measured in feet.

`y` - Vertical distance above the ground, measured in feet.

`a` - Second order coefficient, measured in
`(1)/(ft)`.

`b` - First order coefficient, dimensionless.

`c` - Zero order coefficient, measured in feet.

We can obtain the second-order polynomial associated with the soccer ball by knowing three distinct points and solving the resulting system of linear equations. If we know that
`(x_(1),y_(1)) =(0\,ft, 0\,ft)`,
`(x_(2),y_(2))=(12\,ft, 14\,ft)` and
`(x_(3),y_(3))=(32\,ft, 7\,ft)`, then the system of linear equations is:

`c = 0`

`144\cdot a+12\cdot b +c = 14`

`1024\cdot a +32\cdot b + c = 7`

The solution of this linear system is:

`a = -(91)/(1920)`,
`b = (833)/(480)`,
`c = 0`

Therefore, the function for the height of the ball in terms of its horizontal distance is

`y = -(91)/(1920)\cdot x^(2)+(833)/(480)\cdot x`

In addition, we present a graphic of the given function as attachment.