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]
