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.

(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.

(0,1.563)
; 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.

(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 soccer ball's trajectory represented by the function \( f(x) = -16x^2 + 25x \), we can find the x-intercepts (also known as the roots) by setting the function equal to zero:

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

Factoring out the common terms gives:

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

Setting each factor to zero gives the x-intercepts:

1. \( x = 0 \)
2. \( 25 - 16x = 0 \) → \( 16x = 25 \) → \( x = \frac{25}{16} \) → \( x \approx 1.5625 \)

Thus, the x-intercepts are \( (0, 0) \) and \( \left(1.5625, 0\right) \).

### Interpretation of the x-intercepts:

- **First x-intercept (0, 0)**: This indicates the moment when the soccer ball is kicked from the ground (at time \( x = 0 \)).
- **Second x-intercept \( \left(1.5625, 0\right) \)**: This indicates that the soccer ball will return to the ground approximately **1.5625 seconds** after it has been kicked.

### Graphing the function:

When you graph the function \( f(x) = -16x^2 + 25x \), it will be a downward-opening parabola with its vertex representing the maximum height of the ball. The vertex can be found using the formula \( x = -\frac{b}{2a} \):

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

Substituting this back into the function gives the maximum height:

\[
f\left(\frac{25}{32}\right) = -16\left(\frac{25}{32}\right)^2 + 25\left(\frac{25}{32}\right)
\]

Calculating that gives the peak height.

### Final Answer:

Given the analysis above, the x-intercept \( (1.563, 0) \) fits with our solution:

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