Question

Amir stands on a balcony and throws a ball to his dog, who is at ground level.
The ball's height (in meters above the ground),
\[x\] seconds after Amir threw it, is modeled by
\[h(x)=-(x+1)(x-7)\]
What is the maximum height that the ball will reach?
meters

Answer 1

To find the maximum height of the ball modeled by the function \( h(x) = -(x+1)(x-7) \), we first need to rewrite the function in vertex form, which makes it easier to identify the maximum or minimum value.

We can start by expanding the given function:

\[
h(x) = -(x+1)(x-7)
\]

Expanding the expression:

\[
h(x) = -((x)(x) + (7)(x) - (1)(x) - (7))
\]
\[
= - (x^2 + 6x - 7)
\]
\[
= -x^2 - 6x + 7
\]

Now, we will rewrite this quadratic equation in the vertex form \(a(x - h)^2 + k\) by completing the square. The quadratic function has the form:

\[
h(x) = -1(x^2 + 6x) + 7
\]

To complete the square on \(x^2 + 6x\):

1. Take half of the coefficient of \(x\) (which is \(6\)), square it \((\frac{6}{2})^2 = 9\), and add and subtract this value inside the parentheses.
2. This gives us:

\[
h(x) = -1(x^2 + 6x + 9 - 9) + 7
\]
\[
= -1((x + 3)^2 - 9) + 7
\]
\[
= -1(x + 3)^2 + 9 + 7
\]
\[
= -1(x + 3)^2 + 16
\]

Now, we can see that the vertex of the parabola is at \((-3, 16)\), which represents the maximum height since the parabola opens downwards (indicated by the negative coefficient).

Thus, the maximum height that the ball will reach is:

\[
\boxed{16} \text{ meters}
\]