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)=−16x2+25x
. Graph the function. Identify the x-intercept(s) and interpret its meaning.(1 point)
Responses

(1.25,0)
; The soccer ball will return to the ground 1.25 seconds after it has been kicked.
left parenthesis 1.25 comma 0 right parenthesis ; The soccer ball will return to the ground 1.25 seconds after it has been kicked.

(0,1.563)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 0 comma 1.563 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.

(1.563,0)
; The soccer ball will return to the ground 1.563 seconds after it has been kicked.
left parenthesis 1.563 comma 0 right parenthesis ; The soccer ball will return to the ground 1.563 seconds after it has been kicked.

(0.781,9.766)
; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being kicked.

Answer 1

To analyze the function \( f(x) = -16x^2 + 25x \), which represents the height of the soccer ball as a function of time \( x \), we need to find the \( x \)-intercepts. The \( x \)-intercepts occur where the height \( f(x) \) is zero:

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

Factoring the equation:

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

This gives:

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

Solving for \( x \):

\[
-16x + 25 = 0 \implies 16x = 25 \implies x = \frac{25}{16} \approx 1.5625
\]

Thus, the \( x \)-intercepts are:

1. \( (0, 0) \): This point indicates the moment the ball is kicked from the ground (t = 0).
2. \( \left( \frac{25}{16}, 0 \right) \approx (1.5625, 0) \): This point indicates when the ball returns to the ground.

### Interpretation

The \( x \)-intercept at \( (1.5625, 0) \) means that the soccer ball will return to the ground approximately 1.563 seconds after it has been kicked.

### Conclusion

The correct interpretation is:

**(1.563, 0); The soccer ball will return to the ground 1.563 seconds after it has been kicked.**