To interpret the key features of the quadratic equation \( y = -124x^2 + 2x \) in the context of the baseball game, we need to analyze the components of the equation and what they tell us about the ball's trajectory.
Key Features:
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Vertex: The vertex of a parabola represented by the form \( y = ax^2 + bx + c \) represents the highest or lowest point on the graph. In this case, since \( a < 0 \) (specifically, \( a = -124 \)), we are dealing with a downward-opening parabola, which means the vertex will indicate the highest point the ball reaches.
To find the x-coordinate of the vertex, we use the formula \( x = -\frac{b}{2a} \):
- Here, \( a = -124 \) and \( b = 2 \): \[ x = -\frac{2}{2 \times -124} = \frac{2}{248} = \frac{1}{124} \]
Now, we will substitute \( x = \frac{1}{124} \) back into the equation to find the corresponding \( y \) value (height at the vertex): \[ y = -124\left(\frac{1}{124}\right)^2 + 2\left(\frac{1}{124}\right) \]
Simplifying gives: \[ y = -124 \times \frac{1}{15376} + \frac{2}{124} \] \[ y = -\frac{124}{15376} + \frac{2}{124} = -\frac{124}{15376} + \frac{496}{15376} = \frac{372}{15376} \]
Thus, the vertex (maximum height) of the ball occurs at \( x = \frac{1}{124} \) (approximately \( 0.0081 \) horizontal distance) and \( y \approx 0.0242 \) feet.
Interpretation: The vertex represents the maximum height that the baseball reaches during its trajectory. This is the apex of the ball's flight path before it begins to descend.
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Intercepts: The x-intercepts can be found by setting \( y = 0 \): \[ 0 = -124x^2 + 2x \] Factoring gives: \[ 0 = x(-124x + 2) \] This results in: \[ x = 0 \quad \text{or} \quad -124x + 2 = 0 \Rightarrow x = \frac{2}{124} = \frac{1}{62} \approx 0.0161 \]
Interpretation: The x-intercepts, \( x = 0 \) and \( x \approx 0.0161 \) feet, represent the points where the ball hits the ground. The first intercept at \( 0 \) corresponds to the point where the ball is hit (the contact point), and the second intercept corresponds to the point where the ball lands after its arc.
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Direction: The negative leading coefficient \( (-124) \) indicates that the parabola opens downwards, reinforcing that the ball rises to a peak and then falls back down.
Overall Summary:
In the context of the baseball game, the vertex indicates the maximum height reached by the ball after it is hit, and the x-intercepts show the points where the ball starts (the moment it is hit) and where it lands. The parabola's downward shape represents the ball's trajectory as it ascends and then descends due to gravity.