To interpret the range of the given quadratic equation \( y = -124x^2 + 2x \) in the context of the baseball game, we first need to identify the maximum height of the ball and establish the limits of the range based on the physical scenario presented.
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Finding the Vertex of the Parabola: The equation is in the standard form of a quadratic function, \( y = ax^2 + bx + c \), where \( a = -124 \) and \( b = 2 \).
- The x-coordinate of the vertex, which gives the maximum height (since \( a < 0 \)), can be found using the formula: \[ x = -\frac{b}{2a} = -\frac{2}{2(-124)} = \frac{2}{248} = \frac{1}{124} \]
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Calculating the Maximum Height: Substitute \( x = \frac{1}{124} \) back into the quadratic equation to find the maximum value of \( y \): \[ y = -124\left(\frac{1}{124}\right)^2 + 2 \left(\frac{1}{124}\right) \] \[ = -124 \cdot \frac{1}{15376} + \frac{2}{124} \] \[ = -\frac{124}{15376} + \frac{2}{124} \] \[ = -\frac{124}{15376} + \frac{248}{15376} \] \[ = \frac{124}{15376} = \frac{1}{124} \text{ (when simplifying)} \]
This value simplifies to \( \frac{1}{124} \) or approximately \( 0.0081 \) (in feet).
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Physical Context of the Range: The height \( y \) must be between \( 0 \) (the ground level) and the maximum height calculated. Therefore, if the ball is in the air, its height starts from the ground (0 feet) and reaches its peak at \( \frac{1}{124} \) feet before it comes back down and eventually hits the ground again.
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Conclusion: Thus, in the context of the baseball game, the range of the height \( y \) of the baseball as it travels along its parabolic trajectory can be expressed as: \[ \text{Range: } 0 \leq y \leq \frac{1}{124} \text{ feet} \] This indicates that the ball will not go below ground level and will reach its maximum height of \( \frac{1}{124} \) feet above the ground at its peak in the air.