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Homework Statement
Find the equation of a parabola with a vertex of (1,-3) and focus of (1,-1)?
A parabola is a U-shaped curve that is formed when a straight line intersects a conical surface at a specific angle. It is a type of quadratic function and can be written as y = ax^2 + bx + c, where a, b, and c are constants.
The focus of a parabola is a fixed point inside the curve that is equidistant from each point on the parabola. This point is important because it helps determine the shape and position of the parabola.
The vertex of a parabola is the point where the curve changes direction. It is the highest or lowest point on the parabola depending on the direction it opens. The coordinates of the vertex can be found by using the formula (-b/2a, c - b^2/4a), where a, b, and c are the coefficients in the quadratic function.
To find the equation of a parabola with a given focus and vertex, you can use the formula (x-h)^2 = 4p(y-k), where (h,k) are the coordinates of the vertex and p is the distance from the focus to the vertex. If the parabola opens upwards, p is positive, and if it opens downwards, p is negative.
Yes, a parabola can have a focus and vertex with the same x-coordinate. In this case, the parabola will be a vertical line passing through the focus and vertex. The equation of this parabola would be x = h, where h is the x-coordinate of the focus and vertex.