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 equation of the quadratic function that models the height of the ball \( h(t) \) at time \( t \), we need to use the given points and the standard form of the vertex equation of a parabola.

Given points:
- After 1 second, the ball is 15 feet high: (1, 15)
- After 4 seconds, the ball reaches its maximum height of 42 feet: (4, 42)
- After 7 seconds, it returns to a height of 15 feet: (7, 15)

We are given three key points on the parabolic flight path: the vertex \( (4, 42) \) and two other points \( (1, 15) \) and \( (7, 15) \).

The vertex form of a quadratic equation is:
[tex]\[ h(t) = a(t - h_{\text{shift}})^2 + k \][/tex]
where \( (h_{\text{shift}}, k) \) is the vertex of the parabola. From the problem, we know the vertex \((h_{\text{shift}}, k)\) is \( (4, 42) \).

Thus, we substitute \( h_{\text{shift}} = 4 \) and \( k = 42 \):
[tex]\[ h(t) = a(t - 4)^2 + 42 \][/tex]

Next, we need to find the coefficient \( a \) by plugging in one of the other points, say \( (1, 15) \).

Plugging in \( t = 1 \) and \( h(t) = 15 \):
[tex]\[ 15 = a(1 - 4)^2 + 42 \][/tex]

Simplify the equation:
[tex]\[ 15 = a(-3)^2 + 42 \][/tex]
[tex]\[ 15 = 9a + 42 \][/tex]

Subtract 42 from both sides to solve for \( a \):
[tex]\[ 15 - 42 = 9a \][/tex]
[tex]\[ -27 = 9a \][/tex]
[tex]\[ a = -3 \][/tex]

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

To confirm, we verify with another point, \( (7, 15) \):
[tex]\[ h(7) = -3(7 - 4)^2 + 42 \][/tex]
[tex]\[ h(7) = -3(3)^2 + 42 \][/tex]
[tex]\[ h(7) = -3(9) + 42 \][/tex]
[tex]\[ h(7) = -27 + 42 \][/tex]
[tex]\[ h(7) = 15 \][/tex]

Both points satisfy the equation, confirming our solution is correct. Therefore, the equation modeling the height of the ball is:
[tex]\[ h(t) = -3(t - 4)^2 + 42 \][/tex]

Hence, the correct option is:
[tex]\[ h(t) = -3(t - 4)^2 + 42 \][/tex]