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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.

(0.781, 9.766); 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.
(1.25, 0); The soccer ball will return to the ground 1.25 seconds after it has been kicked.

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To analyze the function \( f(x) = -16x^2 + 25x \), we'll follow these steps: graph the function, identify the x-intercepts, and interpret their meanings.

  1. Graphing the Function: The function \( f(x) = -16x^2 + 25x \) represents a parabolic motion of the soccer ball, opening downwards due to the negative coefficient of \( x^2 \). The graph's vertex represents the maximum height of the ball, while the x-intercepts represent the points where the ball is at ground level (height = 0).

  2. Finding the X-Intercepts: To find the x-intercepts, we set \( f(x) = 0 \):

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

    Factoring out \( x \):

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

    So, the solutions are:

    \[ x = 0 \quad \text{or} \quad -16x + 25 = 0 \Rightarrow -16x = -25 \Rightarrow x = \frac{25}{16} = 1.5625 \]

    Thus, the x-intercepts are:

    • \( x = 0 \)
    • \( x \approx 1.5625 \) (which can be rounded to 1.563 for interpretation).

  3. Interpreting the X-Intercepts:

    • The first x-intercept \( (0, 0) \) means that the soccer ball is kicked from the ground at \( t = 0 \) seconds.
    • The second x-intercept \( (1.563, 0) \) means that the soccer ball returns to the ground approximately 1.563 seconds after it has been kicked.

  4. Finding the Maximum Height: To find the maximum height, we can use the vertex formula \( x = -\frac{b}{2a} \) for a quadratic equation of the form \( ax^2 + bx + c \).

    Here, \( a = -16 \) and \( b = 25 \):

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

    We then substitute \( x \) back into the function \( f(x) \) to find the maximum height:

    \[ f(0.78125) = -16(0.78125)^2 + 25(0.78125) \]

    Calculating stepwise:

    \[ (0.78125)^2 \approx 0.6104 \] \[ -16(0.6104) \approx -9.7664 \] \[ 25(0.78125) = 19.53125 \] \[ f(0.78125) \approx 19.53125 - 9.7664 \approx 9.76485 \]

    Thus, the maximum height reached by the soccer ball is approximately 9.765 feet at \( x \approx 0.781 \) seconds.

Summary of Key Points:

  • The soccer ball reaches a maximum height of approximately 9.765 feet at around \( x \approx 0.781 \) seconds after being kicked;
  • The soccer ball returns to the ground at approximately \( x \approx 1.563 \) seconds after being kicked;
  • The coordinates of the x-intercepts (0, 0) and (1.563, 0) indicate the times at which the ball is at ground level: when kicked and when it lands back.