To interpret the range of the quadratic equation \( y = -124x^2 + 2x \) in the context of a baseball game, we need to analyze the equation.
The given equation represents a downward-opening parabola, as indicated by the negative coefficient of the \( x^2 \) term. The range of a function describes all the possible values of \( y \) (the height of the ball) as \( x \) (the horizontal distance) varies.
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Finding the Vertex: The vertex of the parabola is the highest point (maximum height) the ball reaches during its flight. The vertex can be found using the formula for the x-coordinate of the vertex, \( x = -\frac{b}{2a} \), where \( a = -124 \) and \( b = 2 \).
\[ x = -\frac{2}{2(-124)} = \frac{2}{248} = \frac{1}{124} \]
We substitute this back into the equation to find the corresponding \( y \)-value (the maximum height):
\[ y = -124\left(\frac{1}{124}\right)^2 + 2\left(\frac{1}{124}\right) \] \[ y = -124\left(\frac{1}{15376}\right) + \frac{2}{124} \] \[ y = -\frac{124}{15376} + \frac{2}{124} \]
To find a common denominator and do the calculation, let's convert \(\frac{2}{124}\) into a fraction over 15376:
\[ y = -\frac{124}{15376} + \frac{2 \cdot 124}{15376} = -\frac{124}{15376} + \frac{248}{15376} = \frac{124}{15376} \]
Simplifying \( \frac{124}{15376} \) leads us to:
\[ y = \frac{1}{124} \]
(Note that the simplification is straightforward, since 124 and 15376 share no further common factors in respect to simple reduction. Hence the maximum height is a little higher than this raw formulation.)
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Range Interpretation: Since the ball is hit upward, it will start at ground level, so the minimum height of the ball (when \( y = 0 \)) occurs when \( x = 0 \), and it rises to its maximum height before eventually falling back to \( y = 0 \) again when it lands.
Thus, we conclude:
- The minimum height of the ball is \( y = 0 \) (when the ball is at the ground level).
- The maximum height is \( y = \frac{124}{15376} \) (the highest point reached by the ball before descending).
Therefore, the range of the ball's height (y-values) is:
\[ 0 \leq y \leq \frac{124}{15376} \]
This means the ball travels from a height of 0 feet (ground level) to a maximum height given by the calculated value. So, in the context of the baseball game, the ball's height varies between these two points as it follows a parabolic path.