Question

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 means a is positive

Explanation:

Step1: Analyze the motion of the ball

When a ball is thrown, its height first increases (as it moves up) and then decreases (as it falls back down due to gravity).

Step2: Recall the properties of quadratic graphs

For a quadratic function \(y = ax^{2}+bx + c\), if \(a>0\), the parabola opens upward and has a minimum value. If \(a < 0\), the parabola opens downward and has a maximum value. Since the ball's height has a maximum (it goes up then down), the quadratic graph (which models the height - time relationship \(h(t)=at^{2}+bt + c\), where \(h\) is height and \(t\) is time) must open downward.

Answer:

B. Since the height increases then decreases, the quadratic graph must open downward, not upward, which means a must be negative