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one simple answer is that sometimes it's impossible to describe a curve in the form of y=f(x) since it fails the vertical line test.. the x- and y- coordinates can be each described separately as a function of a parameter "t" for example
I hope that made sense to you
sometimes parametric equations are homogeneous or symmetric .. elegant,that is... while the y = f(x) form is ugly
parametric equation...Think of a curve being traced out over time, sometimes doubling back on itself or crossing itself. Such a curve cannot be described by a function y=f(x). Instead, we will describe our position along the curve at time t by x y = = x(t) y(t) Then x and y are related to each other through their dependence on the parameter t. Example Suppose we trace out a curve according to x y = = t2−4t 3t where t0. The arrow on the curve indicates the direction of increasing time or orientation of the curve. Drag the box along the curve and notice how x and y vary with t. The parameter does not always represent time: Example Consider the parametric equation x y = = 3cos 3sin Here, the parameter represents the polar angle of the position on a circle of radius 3 centered at the origin and oriented counterclockwise. Differentiating Parametric Equations Let x=x(t) and y=y(t). Suppose for the moment that we are able to re-write this as y(t)=f(x(t)). Then dtdy=dxdydtdx by the Chain Rule. Solving for dxdy and assuming dtdx=0, dxdy= dtdx dtdy a formula that holds in general. Example If x=t2−3 and y=t8, then dtdx=2t and dtdy=8t7. So dxdy dx2d2y = = dtdxdtdy=2t8t7=4t6 ddxdxdy= dtdx dtd[dxdy] =2t24t5=12t4 Notes * It is often possible to re-write the parametric equations without the parameter. In the second example, x3=cos, y3=sin. Since cos2+sin2=1, x32+x32=1 . Then x2+y2=9, which is the equation of a circle as expected. When you do eliminate the parameter, always check that you have not introduced extraneous portions of the curve. * Every curve has infinitely many parametrizations, amounting to different scales for the parameter. For example, x y = = 3cos2 3sin2 traces out the circle from the second example twice as "quickly," completing a full revolution in rather than 2 units of . * Every equation y=f(x) may be re-written in parametric form by letting x=t, y=f(t). Key Concepts A curve in the xy-plane may be described by a pair of parametric equations x=x(t) y=y(t) where x and y are related through their dependence on t. This is particularly useful when neither x nor y is a function of the other. The derivative of y with respect to x (in terms of the parameter t) is given by dxdy=dtdxdtdy