The function \( f(x) = -16x^2 + 38x \) models the height of the soccer ball (in feet) as a function of time (in seconds) after being kicked. Here's a breakdown of the components of the function and its meaning:
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Quadratic Equation: The function is a quadratic equation representing a parabolic trajectory, typical for projectile motion due to the influence of gravity.
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Coefficients:
- The term \(-16x^2\) represents the effect of gravitational acceleration (in feet per second squared, on Earth, this is approximately \( -32 \) feet/s² divided by 2).
- The term \(38x\) indicates the initial upward velocity of the ball, which is \( 38 \) feet/second.
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Meaning of the Function:
- \(x\): This variable represents time in seconds after the ball has been kicked.
- \(f(x)\): This variable represents the height of the ball at time \(x\).
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Parabolic Motion: The curve opens downward due to the negative coefficient of the \(x^2\) term, indicating that the ball will rise to a peak height (the maximum value of the function) before descending back to the ground (where the height is \( f(x) = 0\)).
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Finding Key Features:
- Vertex: The vertex of the parabola represents the maximum height the ball reaches.
- Roots: The points where \( f(x) = 0 \) are the times when the ball hits the ground. You would solve the equation to find these values.
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Real-World Implications: This function can be used to determine how long the ball will be in the air and what maximum height it will reach, both important for analyzing the effectiveness of the kick.
In conclusion, \( f(x) = -16x^2 + 38x \) effectively models the height of a soccer ball over time after being kicked, taking into account the initial velocity and the force of gravity acting on the ball.
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