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

Answer:

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

$y = -\frac{91}{1920}\cdot x^{2}+\frac{833}{480}\cdot x$

Step-by-step 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 $\frac{1}{ft}$.

$b$ - 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:

$a = -\frac{91}{1920}$, $b = \frac{833}{480}$, $c = 0$

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

$y = -\frac{91}{1920}\cdot x^{2}+\frac{833}{480}\cdot x$

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