To represent the scenario of Lorraine hitting the golf ball, we can formulate it as a quadratic equation.
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Equation: The ball's flight can be modeled by a quadratic equation in the form \( y = ax^2 + bx + c \), where \( y \) represents the height of the ball and \( x \) represents the horizontal distance from Lorraine.
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Landing Point: The ball lands 60 feet away from her, which means this point can be represented as \( (60, 0) \). This is the x-intercept of the parabola.
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Axis of Symmetry: The axis of symmetry for a parabola described by a quadratic equation is located at the vertex and can be found at \( x = \frac{p + q}{2} \), where \( p \) and \( q \) are the x-intercept points. Since the ball lands 60 feet away and starts at 0 feet away, the distance from the launch point to the landing point is 60 feet. Thus, the halfway point for the axis of symmetry is located at: \[ x = \frac{0 + 60}{2} = 30 \text{ feet} \]
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Vertex: Given that the highest point of the ball's trajectory is 30 feet in the air, the vertex of the parabola is at \( (30, 30) \).
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Starting Point: The ball starts at 0 feet away from Lorraine, so another point on the graph would be \( (0, 0) \).
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Missing Value: Using the information given, the only value that is not explicitly stated and needs to be found is the coefficient \( a \) in the quadratic equation. This coefficient determines how "wide" or "narrow" the parabola is.
To summarize:
- The scenario is best represented by a quadratic equation.
- The ball lands 60 feet away represents the \( x \)-intercept.
- Half of this distance gives the axis of symmetry at 30 feet.
- The vertex is at \( (30, 30) \).
- The ball starts at \( (0, 0) \).
- The missing value to be determined is the coefficient \( a \).
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