To analyze the function \( f(x) = -16x^2 + 25x \), which describes the height of a soccer ball over time as it is kicked upwards, we can follow these steps:
1. Finding the x-intercepts:
To find the x-intercepts, we need to set \( f(x) = 0 \):
\[ -16x^2 + 25x = 0 \]
Factoring out \( x \):
\[ x(25 - 16x) = 0 \]
This gives us two solutions:
- \( x = 0 \)
- \( 25 - 16x = 0 \)
\[ 16x = 25 \quad \Rightarrow \quad x = \frac{25}{16} \approx 1.5625 \]
2. Interpreting the x-intercepts:
The x-intercepts are values of \( x \) when the height \( f(x) \) is zero, which means the soccer ball is on the ground. The two intercepts are:
- \( (0, 0) \): This point represents the moment the ball is kicked (time = 0 seconds).
- \( \left( \frac{25}{16}, 0 \right) \approx (1.563, 0) \): This point represents the time when the ball returns to the ground after being kicked.
3. Finding the maximum height:
To find the maximum height of the function, we can use the vertex formula for a parabola. The vertex \( x \) coordinate can be found using \( x = -\frac{b}{2a} \) where \( a = -16 \) and \( b = 25 \):
\[ x = -\frac{25}{2 \times -16} = \frac{25}{32} \approx 0.78125 \]
Next, we evaluate \( f \left( \frac{25}{32} \right) \) to find the maximum height:
\[ f(0.78125) = -16(0.78125)^2 + 25(0.78125) \]
Calculating \( f(0.78125) \):
\[ = -16(0.6103515625) + 19.53125 \] \[ \approx -9.685625 + 19.53125 = 9.845625 \text{ feet} \]
Summary of Findings:
- The soccer ball will return to the ground approximately 1.563 seconds after being kicked (indicated by the x-intercept \( (1.563, 0) \)).
- The maximum height of approximately 9.85 feet occurs at about 0.781 seconds after being kicked.
Conclusion:
The relevant x-intercepts and their meaning are:
- (0, 0): The moment the ball is kicked.
- (1.563, 0): The time when the ball hits the ground again after being kicked.
The maximum height is around 9.85 feet reached at approximately 0.781 seconds after the kick, indicating the peak point of the ball's trajectory.
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