To determine which function models the graph of the height reached by the ball after being hit, we need to note a few characteristics of a quadratic function that models projectile motion:
- The function should be concave down, meaning that the coefficient of the squared term must be negative.
- The vertex of the parabola typically represents the maximum height, which would be at or near the point where the ball reaches its peak.
- The general form of the function can be expressed as \(h(x) = a(x - h)^2 + k\), where \((h, k)\) is the vertex. If \(a < 0\), the parabola opens downwards.
Now, examining the provided options:
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The first function \(h(x) = \frac{18}{9,245}(x-215)^2 + 90\) is upward-opening (\(a > 0\)), which does not fit our expectations for a height function reaching a maximum.
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The second function \(h(x) = -\frac{43}{1,620}(x-90)^2 + 215\) has \(a < 0\), indicating it is downward-opening, which fits the situation. The vertex is at \( (90, 215) \), indicating that the maximum height of the ball is 215 feet at a horizontal distance of 90 feet, which makes sense in context.
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The third function \(h(x) = -\frac{18}{9,245}(x-215)^2 + 90\) is again downward-opening, but the maximum height of 90 feet is very low for a baseball. It doesn't represent a likely scenario for a home run hit.
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The fourth function \(h(x) = \frac{43}{1,620}(x-90)^2 + 215\) is upward-opening (\(a > 0\)) and does not fit either.
Thus, the only function that models the height reached by the ball accurately is the second option:
The height reached by the ball, \(h\left(x\right)\), is given by the equation \(h\left(x\right)=-\frac{43}{1,620}\left(x-90\right)^2+215\), where \(x\) represents the horizontal distance.