Parametric Equations of Parabolas

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Parametric equations of a parabola represent the coordinates of points on the parabolic curve using two parameters (usually denoted as t or ), Instead of expressing the curve in terms of x and y directly as in the Cartesian coordinate system, parametric equations use two separate equations for x and y in terms of the parameter t. The general form of parametric equations for a parabola is: x = f(t) y = g(t) where f(t) and g(t) are functions that define the x and y coordinates respectively, in terms of the parameter t. For a parabola, the parametric equations can be defined using either the vertex form or the standard form of a parabola. 1. Vertex Form Parametric Equations: For a parabola with its vertex at (h,k), the parametric equations are often written as: x = h + at2 y = k + bt Where a determines the direction and width of the parabola, and b controls the vertical shift of the parabola. 2. Standard Form Parametric Equations: The standard form of a parabola is given by: y = ax2 + bx + c. To represent it in parametric form, we can use the following equations: x = t y = at2 +bt + c In this case, the parameter f directly corresponds to the x coordinate, while the y coordinate is expressed as a function of t using the parabolic equation. Parametric equations of a parabola are particularly useful in computer graphics, physics, and engineering applications, where motion along a curved path or trajectories of objects can be described efficiently using parameterization. Additionally, parametric equations allow for a more straightforward representation of parabolas when considering more complex curves, such as rotated or translated parabolic shapes.