Plane CurvesUntil now, you have been representing a graph by a single equation involving two variables. In this section, you will study situations in which three variables are used to represent a curve in a plane. Consider the path followed by an object that is propelled into the air at an angle of 45°. If the initial velocity of the object is 48 feet per second, the object travels the parabolic path given by Rectangular Equation However, this equation does not tell the whole story. Although it tells you where the object has been, it doesn’t tell you when the object was at a given point . To determine the time, we can introduce a third variable, t, called a parameter.By Writing both xand yas functions of t, you obtain the parametric equations. Parametric Equation for x Parametric Equation for yFrom this information, you can determine that at time t= 0, the object is at the point (0, 0). Similarly, at time t= 1, the object is at the point and so on. For this problem, xand yare continuous functions of t, and the resulting path is called the plane curve.Example 1: Sketching a Curve. Sketch the curve described by the parametric equations t x y 272xyx=-+(),x y242xt=×216242ytt=-+×(242,24216)-24 and ,23.2txtyt=-=-££2-1-0123DEFINITION OF A PLANE CURVE If fand g are continuous functions of ton an interval I, then the equations and are called parametric equations and tis called the parameter. The set of pointsobtained as tvaries over the interval Iis called the graphof the parametric equations. Taken together, the parametric equations and the graph are called a plane curve, denoted by C. When sketching a curve by hand represented by parametric equations, you use increasing values of t. Thus the curve will be traced over a specific direction. This is called the orientationof the curve. You use arrowsto show the orientation. ( )xf t=( )yg t=( ,)x y-4-3-2-112345-5-4-3-2-11234xy
Example 2: Sketching a Curve. Sketch the curve described by the parametric equations t x y Many times when a parametric equation is given, we wish only to sketch the general shape of the plane curve. In that case, we wish to eliminate the parameterto create a rectangular equation in the form of . The technique to accomplish this is to solve for the parameter in one of the parametric equations (choosing the easiest to do so) and then replacing the result in the other equation. Eliminating the Parameter Example 3: Adjusting the Domain After Eliminating the Parameter. Sketch the curve described by the parametric equations by eliminating the parameter and adjusting the domain of the resulting rectangular equation. Angles as Parameters Example 4: Using Trigonometry to Eliminate a Parameter. Sketch the curve represented by by eliminating the parameter and finding the corresponding rectangular equation.