To determine the best function that models the height of a baseball as it travels through the air and forms a concave curve, we need to consider the general form of a quadratic equation representing the trajectory of the ball:
\[ h(x) = a(x - h)^2 + k\]
Here, \( (h, k) \) represents the vertex of the parabola, where \( h \) is the horizontal distance at which the maximum height occurs and \( k \) is the maximum height reached.
Given that the curve is concave down (the height of the ball reaches a maximum), the coefficient \( a \) must be negative.
Analyzing the Options:
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Option 1: \( h(x) = -\frac{431}{1,620}(x - 90)^2 + 215 \)
- This option has a negative leading coefficient, which is appropriate for a concave down curve.
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Option 2: \( h(x) = -\frac{18}{9,245}(x - 215)^2 + 90 \)
- This also has a negative leading coefficient.
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Option 3: \( h(x) = \frac{431}{1,620}(x - 90)^2 + 215 \)
- This has a positive leading coefficient, so it cannot represent a concave down trajectory.
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Option 4: \( h(x) = 189,245(x - 215)^2 + 90 \)
- This also has a positive leading coefficient, indicating an upward opening parabola.
Conclusion:
The equations in Options 1 and 2 are viable because they both have negative leading coefficients. However, it’s important to note that the values of \( a \) impact how wide or steep the curve is and the chosen vertex parts \( (h, k) \) impact the effect on horizontal and vertical position.
Given that we are trying to model a baseball trajectory, Option 1 seems more realistic with reasonably high numbers which indicate a typical baseball height, providing a maximum height of 215 feet at a distance of 90 feet.
Thus, the best answer is:
The height reached by the ball, h(x) is given by the equation \( h(x) = -\frac{431}{1,620}(x - 90)^2 + 215 \).