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 (1.563,0) ; 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. left parenthesis 0.781 comma 9.766 right parenthesis ; The soccer ball will reach a maximum height of 9.766 feet 0.781 seconds after being 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. (1.25,0) ; 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.

Answer 1

To understand the function \( f(x) = -16x^2 + 25x \) that represents the height of the soccer ball over time, we need to find the x-intercepts and interpret their meanings.

Finding the x-intercepts

To find the x-intercepts of the function, we set \( f(x) = 0 \):

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

Factoring the equation gives:

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

Setting each factor to zero yields:

1. \( x = 0 \)
2. \( 25 - 16x = 0 \)

Solving \( 25 - 16x = 0 \):

\[ 16x = 25 \ x = \frac{25}{16} \ x = 1.5625 \]

Interpretation of the x-intercepts

The x-intercepts are at \( (0, 0) \) and approximately \( (1.563, 0) \).

1. First x-intercept (0, 0): This point indicates that the soccer ball starts at the ground (height = 0) when \( x = 0 \) seconds—that is, at the moment of kicking.

2. Second x-intercept (1.563, 0): This point indicates that the soccer ball will return to the ground (height = 0) approximately 1.563 seconds after it has been kicked.

Conclusion

The correct interpretation of the x-intercept \( (1.563, 0) \) is:

• The soccer ball will return to the ground 1.563 seconds after it has been kicked.

Therefore, the response that matches this interpretation is:

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