Question

A soccer ball is kicked from the ground at an upward velocity of 25 feet per second. This situation can be represented by the function
f(x) = - 16x ^ 2 + 25x Graph the function. Identify the x-intercept(s) and interpret its meaning.

(0.781, 9.766); The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.
(1.563, 0) The soccer ball will return to the ground 1.563 seconds after it has been kicked.
(1.25, 0); The soccer ball will return to the ground 1.25 seconds after it has been kicked.

Answer 1

To analyze the function \( f(x) = -16x^2 + 25x \), we'll follow these steps: graph the function, identify the x-intercepts, and interpret their meanings.

1. **Graphing the Function**:
The function \( f(x) = -16x^2 + 25x \) represents a parabolic motion of the soccer ball, opening downwards due to the negative coefficient of \( x^2 \). The graph's vertex represents the maximum height of the ball, while the x-intercepts represent the points where the ball is at ground level (height = 0).

2. **Finding the X-Intercepts**:
To find the x-intercepts, we set \( f(x) = 0 \):

\[
-16x^2 + 25x = 0
\]

Factoring out \( x \):

\[
x(-16x + 25) = 0
\]

So, the solutions are:

\[
x = 0 \quad \text{or} \quad -16x + 25 = 0 \Rightarrow -16x = -25 \Rightarrow x = \frac{25}{16} = 1.5625
\]

Thus, the x-intercepts are:
- \( x = 0 \)
- \( x \approx 1.5625 \) (which can be rounded to 1.563 for interpretation).

3. **Interpreting the X-Intercepts**:
- The first x-intercept \( (0, 0) \) means that the soccer ball is kicked from the ground at \( t = 0 \) seconds.
- The second x-intercept \( (1.563, 0) \) means that the soccer ball returns to the ground approximately 1.563 seconds after it has been kicked.

4. **Finding the Maximum Height**:
To find the maximum height, we can use the vertex formula \( x = -\frac{b}{2a} \) for a quadratic equation of the form \( ax^2 + bx + c \).

Here, \( a = -16 \) and \( b = 25 \):

\[
x = -\frac{25}{2 \cdot -16} = \frac{25}{32} \approx 0.78125
\]

We then substitute \( x \) back into the function \( f(x) \) to find the maximum height:

\[
f(0.78125) = -16(0.78125)^2 + 25(0.78125)
\]

Calculating stepwise:

\[
(0.78125)^2 \approx 0.6104
\]
\[
-16(0.6104) \approx -9.7664
\]
\[
25(0.78125) = 19.53125
\]
\[
f(0.78125) \approx 19.53125 - 9.7664 \approx 9.76485
\]

Thus, the maximum height reached by the soccer ball is approximately 9.765 feet at \( x \approx 0.781 \) seconds.

### Summary of Key Points:
- The soccer ball reaches a maximum height of approximately 9.765 feet at around \( x \approx 0.781 \) seconds after being kicked;
- The soccer ball returns to the ground at approximately \( x \approx 1.563 \) seconds after being kicked;
- The coordinates of the x-intercepts (0, 0) and (1.563, 0) indicate the times at which the ball is at ground level: when kicked and when it lands back.