Question

a golfer hits a golf ball. the graph of ( h(t) ) below shows the height of the ball over time. interpret the meaning of the vertex in this situation. the golf ball reaches a maximum height of feet after seconds. interpret the meaning of the horizontal intercepts in this situation. when the golf ball was hit, it was feet above the ground. the ball lands after seconds. practical domain ( leq t leq ) seconds practical range ( leq h(t) leq ) feet

Explanation:

1. Analyze the vertex of the parabola

The vertex of a parabola \(y = ax^{2}+bx + c\) (in this case \(H(t)\) is a quadratic - like function representing the height of the ball over time) gives the maximum or minimum value. Since the parabola opens downwards (the coefficient of the \(t^{2}\) term is negative as the ball goes up and then comes down), the vertex gives the maximum height.
Looking at the graph, the \(y\) - coordinate (height) of the vertex is \(120\) feet and the \(x\) - coordinate (time) is \(4\) seconds.

2. Analyze the horizontal intercepts

The horizontal intercepts of the graph \(H(t)\) (where \(H(t)=0\)):

• When \(t = 0\) (the ball is hit), \(H(0)=0\) (the left - hand horizontal intercept).
• The right - hand horizontal intercept occurs when \(t = 8\) seconds (the ball lands).

3. Determine the practical domain and range

• Practical domain: The domain of a function is the set of all possible input values. For the height of the ball over time, the ball is in the air from \(t = 0\) (when it is hit) to \(t = 8\) (when it lands). So, \(0\leq t\leq8\).
• Practical range: The range of a function is the set of all possible output values. The minimum height of the ball is \(0\) (when it is on the ground, at \(t = 0\) and \(t = 8\)) and the maximum height is \(120\) feet (from the vertex). So, \(0\leq H(t)\leq120\).

Answer:

• The golf ball reaches a maximum height of \(120\) feet after \(4\) seconds.
• When the golf ball was hit, it was \(0\) feet above the ground. The ball lands after \(8\) seconds.
• Practical Domain: \(0\leq t\leq8\) seconds
• Practical Range: \(0\leq H(t)\leq120\) feet