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
𝑓
(
𝑥
)
=
−
16
𝑥
2
+
25
𝑥
. Graph the function. Identify the x-intercept(s) and interpret its meaning.(1 point)
(1.563,0)
(
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)
(
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.
(1.25,0)
(
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.

(0.781,9.766)
(
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 \( f(x) = -16x^2 + 25x \) and determine the x-intercepts and maximum height, we'll start by finding the x-intercepts, which occur when \( f(x) = 0 \).

### Step 1: Finding the x-intercepts
Set the function equal to zero:
\[
-16x^2 + 25x = 0
\]
Factor out \( x \):
\[
x(-16x + 25) = 0
\]
Setting each factor to zero gives:
1. \( x = 0 \)
2. \( -16x + 25 = 0 \)
\[
16x = 25 \quad \Rightarrow \quad x = \frac{25}{16} = 1.5625
\]

### Summary of x-intercepts
Thus, the x-intercepts are:
1. \( (0, 0) \)
2. \( (1.5625, 0) \)

### Interpretation
- \( (0, 0) \): This point represents the soccer ball being kicked from the ground at time \( x = 0 \) seconds.
- \( (1.5625, 0) \): This point indicates that the soccer ball returns to the ground after approximately 1.563 seconds.

### Step 2: Finding the maximum height
To find the maximum height, we can calculate the vertex of the parabola given by \( f(x) = -16x^2 + 25x \). The x-coordinate of the vertex of a parabola in standard form \( ax^2 + bx + c \) can be found using:
\[
x_v = -\frac{b}{2a}
\]
In our function:
- \( a = -16 \)
- \( b = 25 \)

Calculating the vertex:
\[
x_v = -\frac{25}{2(-16)} = \frac{25}{32} \approx 0.78125
\]

Now, substituting \( x_v \) back into the function to find the maximum height:
\[
f(0.78125) = -16(0.78125)^2 + 25(0.78125)
\]
Calculating \( f(0.78125) \):
\[
f(0.78125) = -16(0.6103515625) + 19.53125 \approx -9.645625 + 19.53125 = 9.885625 \approx 9.89 \text{ feet}
\]

### Summary of maximum height
Thus, at approximately \( (0.78125, 9.89) \), the soccer ball reaches its maximum height of about 9.89 feet approximately 0.78 seconds after being kicked.

### Conclusion
- The x-intercepts are:
- \( (0, 0) \): The soccer ball is on the ground before it's kicked.
- \( (1.5625, 0) \): The soccer ball returns to the ground 1.563 seconds after it is kicked.

- The maximum height is approximately 9.89 feet, reached at \( (0.78125, 9.89) \) seconds after being kicked.