[size=150]Parametric equations of a parabola represent the coordinates of points on the parabolic curve using two parameters (usually denoted as [i]t[/i] or [math]\Theta[/math]), Instead of expressing the curve in terms of [i]x[/i] and [i]y[/i] directly as in the Cartesian coordinate system, parametric equations use two separate equations for [i]x[/i] and y in terms of the parameter [i]t.[/i][br][br]The general form of parametric equations for a parabola is:[br][br]x = f(t) [br]y = g(t)[br][br]where f(t) and g(t) are functions that define the [i]x[/i] and [i]y[/i] coordinates respectively, in terms of the parameter [i]t.[br][/i][br]For a parabola, the parametric equations can be defined using either the vertex form or the standard form of a parabola.[br][br]1. Vertex Form Parametric Equations:[br]For a parabola with its vertex at (h,k), the parametric equations are often written as:[br][br] x = h + at[sup]2[/sup][br] y = k + bt[br][br]Where [i]a[/i] determines the direction and width of the parabola, and [i]b[/i] controls the vertical shift of the parabola.[br][br]2. Standard Form Parametric Equations:[br]The standard form of a parabola is given by: y = ax[sup]2[/sup] + bx + c. To represent it in parametric form, we can use the following equations:[br][br] x = t [br] y = at[sup]2 [/sup]+bt + c [br][br]In this case, the parameter [i]f [/i]directly corresponds to the [i]x[/i] coordinate, while the [i]y[/i] coordinate is expressed as a function of [i]t [/i]using the parabolic equation.[br][br]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.[/size]