Here are the essential concepts you must grasp in order to answer the question correctly.
Standard Form of a Parabola
The standard form of a parabola that opens horizontally is given by the equation (y - k)² = 4p(x - h), where (h, k) is the vertex and p is the distance from the vertex to the focus or directrix. This form allows for easy identification of the parabola's orientation and key features.
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Focus and Directrix
In the context of parabolas, the focus is a fixed point from which distances to points on the parabola are measured, while the directrix is a line that is equidistant from the focus. The parabola is defined as the set of points that are equidistant from both the focus and the directrix, which is crucial for deriving its equation.
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Vertex of a Parabola
The vertex of a parabola is the midpoint between the focus and the directrix. It represents the point at which the parabola changes direction. For the given focus (3, 2) and directrix x = -1, the vertex can be calculated as the average of the x-coordinates of the focus and the directrix, which is essential for writing the standard form of the parabola.
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