Question

Emit threw a football during practice. The function h(x)= -1/40x2 + x +6



represents the path of the football.

Emit wants to rewrite the function in a form that reveals its vertex.

Complete the form for h(x) that reveals its vertex, using parentheses appropriately, then state the maximum height the football reached.

The function can be rewritten as h(x)= -1/40 __to reveal its vertex.
The maximum height the football reached was __
feet.

Answer 1

To rewrite the function \( h(x) = -\frac{1}{40}x^2 + x + 6 \) in vertex form, we need to complete the square.

1. Start with the function:
\[
h(x) = -\frac{1}{40}x^2 + x + 6
\]

2. Factor out \(-\frac{1}{40}\) from the quadratic and linear terms:
\[
h(x) = -\frac{1}{40}(x^2 - 40x) + 6
\]

3. Now, complete the square inside the parentheses. To do that, we take half of the coefficient of \(x\) (which is \(-40\)), square it, and then add and subtract that value:
- Half of \(-40\) is \(-20\), and squaring it gives \(400\).

So we rewrite:
\[
h(x) = -\frac{1}{40}(x^2 - 40x + 400 - 400) + 6
\]

4. This simplifies to:
\[
h(x) = -\frac{1}{40}((x - 20)^2 - 400) + 6
\]

5. Distributing inside the parentheses gives:
\[
h(x) = -\frac{1}{40}(x - 20)^2 + 10 + 6
\]

6. Combine like terms:
\[
h(x) = -\frac{1}{40}(x - 20)^2 + 16
\]

Now, we have the vertex form of the equation, which is:
\[
h(x) = -\frac{1}{40}(x - 20)^2 + 16
\]

Thus, **the function can be rewritten as**:
\[
h(x) = -\frac{1}{40}(x - 20)^2 + 16
\]

**The maximum height the football reached was**:
\[
16 \text{ feet}.
\]

So, filling in the blanks:

The function can be rewritten as \( h(x) = -\frac{1}{40}(x - 20)^2 + 16 \) to reveal its vertex. The maximum height the football reached was \( 16 \) feet.