CURVES DEFINED BY PARAMETRIC EQUATIONSImagine that a particle moves along the curveCshown in Figure 1. It is impossible todescribeCby an equation of the formbecauseCfails the Vertical Line Test. Butthex- andy-coordinates of the particle are functions of time and so we can writeand. Such a pair of equations is often a convenient way of describing a curve andgives rise to the following definition.Suppose thatandare both given as functions of a third variable(called aparam-eter) by the equations(calledparametric equations). Each value ofdetermines a point, which we canplot in a coordinate plane. Asvaries, the pointvaries and traces outa curve, which we call aparametric curve. The parametertdoes not necessarily repre-sent time and, in fact, we could use a letter other thantfor the parameter. But in manyapplications of parametric curves,tdoes denote time and therefore we can interpretas the position of a particle at timet.EXAMPLE 1Sketch and identify the curve defined by the parametric equationsSOLUTIONEach value ofgives a point on the curve, as shown in the table. For instance, if, then,and so the corresponding point is. In Figure 2 we plot thepointsdetermined by several values of the parameter and we join them to producea curve.A particle whose position is given by the parametric equations moves along the curvein the direction of the arrows asincreases. Notice that the consecutive points marked onthe curve appear at equal time intervals but not at equal distances. That is because theparticle slows down and then speeds up asincreases.It appears from Figure 2 that the curve traced out by the particle may be a parabola.This can be confirmed by eliminating the parameteras follows. We obtainfrom the second equation and substitute into the first equation. This givesand so the curve represented by the given parametric equations is the parabola.Mxy24y3xt22ty122y1y24y3ty1tttFIGURE 20t=0t=1t=2t=3t=4t=_1t=_2(0,1)yx8x,y0, 1y1x0t0tyt1xt22tx,yf t,ttCx,yf t,tttx,ytyttxf ttyxyttxf tyf x10.1txy281130001112203334485NThis equation inanddescribeswheretheparticle has been, but it doesn’t tell uswhenthe