Dominique throws a softball from the outfield. After 1 second, the ball is 15 feet high. After 4 seconds, the ball reaches its maximum height of 42 feet. After 7 seconds, it returns to a height of 15 feet.

What is the equation of the quadratic function that models the height of the ball \( h(t) \) at time \( t \)?

A. \( h(t) = -2(t - 4)^2 + 42 \)
B. \( h(t) = 2(t + 4)^2 + 42 \)
C. \( h(t) = -3(t - 4)^2 + 42 \)
D. [tex]\( h(t) = 3(t + 4)^2 + 42 \)[/tex]



Answer :

To determine the correct quadratic function that models the height of the ball, let's carefully examine the given conditions and evaluate each formula step-by-step.

1. The ball's height values at specific times are:
- After \(1\) second, \(h(1) = 15\) feet.
- After \(4\) seconds, \(h(4) = 42\) feet (the maximum height).
- After \(7\) seconds, \(h(7) = 15\) feet.

Given the four candidate functions:
1. \(h(t) = -2(t - 4)^2 + 42\)
2. \(h(t) = 2(t + 4)^2 + 42\)
3. \(h(t) = -3(t - 4)^2 + 42\)
4. \(h(t) = 3(t + 4)^2 + 42\)

Let's check each to identify which one satisfies all three conditions.
### Evaluating \(h(t) = -2(t - 4)^2 + 42\)
1. \(h(1)\):
[tex]\[ h(1) = -2(1 - 4)^2 + 42 = -2(-3)^2 + 42 = -2 \cdot 9 + 42 = -18 + 42 = 24 \][/tex]
\(h(1)\) should be 15, but here we got 24. This function is incorrect.

### Evaluating \(h(t) = 2(t + 4)^2 + 42\)
2. \(h(1)\):
[tex]\[ h(1) = 2(1 + 4)^2 + 42 = 2(5)^2 + 42 = 2 \cdot 25 + 42 = 50 + 42 = 92 \][/tex]
\(h(1)\) should be 15, but here we got 92. This function is incorrect.

### Evaluating \(h(t) = -3(t - 4)^2 + 42\)
3. \(h(1)\):
[tex]\[ h(1) = -3(1 - 4)^2 + 42 = -3(-3)^2 + 42 = -3 \cdot 9 + 42 = -27 + 42 = 15 \][/tex]
\(h(1) = 15\), which matches the given height.

\(h(4)\):
[tex]\[ h(4) = -3(4 - 4)^2 + 42 = -3(0)^2 + 42 = 0 + 42 = 42 \][/tex]
\(h(4) = 42\), which matches the given maximum height.

\(h(7)\):
[tex]\[ h(7) = -3(7 - 4)^2 + 42 = -3(3)^2 + 42 = -3 \cdot 9 + 42 = -27 + 42 = 15 \][/tex]
\(h(7) = 15\), which matches the given height.

This function satisfies all three conditions.

### Evaluating \(h(t) = 3(t + 4)^2 + 42\)
4. \(h(1)\):
[tex]\[ h(1) = 3(1 + 4)^2 + 42 = 3(5)^2 + 42 = 3 \cdot 25 + 42 = 75 + 42 = 117 \][/tex]
\(h(1)\) should be 15, but here we got 117. This function is incorrect.

Thus, the correct quadratic function modeling the height of the ball is:
[tex]\[ h(t) = -3(t - 4)^2 + 42 \][/tex]