Question

a ball is thrown upward and outward from a height of 5 feet. the table shows four measurements indicating the balls height at various horizontal distances from where it was thrown. a graphing calculator displays a quadratic function that models the balls height, y, in feet, in terms of its horizontal distance, x, in feet. answer parts (a)-(c) below.

x, balls horizontal distance (feet)	y, balls height (feet)
1	7.7
3	5
4	3.1

quadreg
y = ax²+bx + c
a=-0.8
b = 2.4
c = 5.4
a. explain why a quadratic function was used to model the data
a. the height increases then decreases, which looks like a quadratic function
b. the height decreases then increases, which looks like a quadratic function
c. the height decreases quickly at first, but then decreases more slowly, which looks like a quadratic function
d. the height starts decreasing more and more rapidly, which looks like a quadratic function
in the quadratic regression screen shown in the problem statement, why is the value of the coefficient a negative?
a. since the height increases then decreases, the quadratic graph must open upward, not downward, which means a must be negative
b. since the height increases then decreases, the quadratic graph must open downward, not upward, which means a must be negative
c. since the height of the ball starts at the height of the person throwing the ball, the quadratic graph must be shifted up, which makes a negative

Explanation:

For the first part, a quadratic function is used because the ball's height first increases (as it is thrown upward) and then decreases (due to gravity pulling it down), which is characteristic of a quadratic - shaped curve. For the second part, in a quadratic function \(y = ax^{2}+bx + c\), when the graph opens downward (the height of the ball increases then decreases), the coefficient \(a\) of the \(x^{2}\) term is negative.

Answer:

a. A. The height increases then decreases, which looks like a quadratic function
b. B. Since the height increases then decreases, the quadratic graph must open downward, not upward, which means a must be negative