Chapter 1 Topics in Analytic Geometry

•••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 16OrientationThe direction inwhich thegraphof apair of parametric equations is traces asthe parameterincreasesiscalledthedirection of increasing parameter orsometimestheorientationimposed on the curve by the equation. Thus, we make a distinction between a curve, whichis the set of points, and a parametric curve, which is a curve with an orientation. Forexample, we saw in Example 3.28 that the circle represented parametrically by (2.1) istraced counterclockwise as t increases and hence has counterclockwise orientation.To obtain parametric equation for the unit circle with clockwise orientation, we canreplace t by −t in (2.1). This yieldsx = cos(−t) = cost, y = sin(−t) = −sint (0 ≤ t ≤ 2π)Here, the circle is traced clockwise by a point that starts at (1,0) when t = 0 and completesone full revolution when t = 2π.y1xty(x,y)(0,1)xExample 2.3 Graph the parametric curvex = 2t−3, y = 6t−7by eliminating the parameter, and indicate the orientation on the graph.Solution .........Expressing Ordinary Functions ParametricallyAn equation y = f(x) can be expressed in parametric form by introducing the parametert = x; this yields the parametric equationsx = t, y = f(t)For example, the portion of the curve y = cosx over the interval [−2π,2π] can be expressedparametrically asx = t, y = cost (−2π ≤ t ≤ 2π)