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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.(1 point)

Responses

(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.

(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.

(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.

1 answer

To analyze the function \( f(x) = -16x^2 + 25x \), which models the height of the soccer ball over time (with \( x \) representing time in seconds and \( f(x) \) representing the height in feet), we need to find the x-intercepts. The x-intercepts represent the moments when the ball is at ground level (height = 0).

To find the x-intercepts, we set the function equal to zero:

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

Factoring out \( x \):

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

This gives us two solutions:

  1. \( x = 0 \) (the moment the ball is kicked)
  2. From \( -16x + 25 = 0 \):

\[ -16x = -25 \implies x = \frac{25}{16} \approx 1.563 \]

Thus, the x-intercepts are \( (0, 0) \) and \( \left(\frac{25}{16}, 0\right) \).

Now let's interpret the second x-intercept:

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

Therefore, the relevant response is:

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