736 CHAPTER 10 Analytic Geometry 1 Graph Parametric Equations by Hand Parametric equations are particularly useful in describing movement along a plane curve. Suppose that a plane curve is defined by the parametric equations x x t y y t a t b ( ) ( ) = = ≤ ≤ where each function is defined over the interval a t b. ≤ ≤ For a given value of t, the values of x x t( ) = and y y t( ) = determine a point x y , ( ) on the curve. In fact, as t varies over the interval from t a = to t b, = successive values of t determine the direction of the movement along the curve.That is, the curve is traced out in a certain direction by the corresponding succession of points x y , . ( ) See Figure 62.The arrows show the direction, or orientation , along the curve as t varies from a to b. Figure 62 Plane curve x A 5 (x(a), y(a)) B 5 (x(b), y(b)) P 5 (x(t), y(t)) t 5 a t 5 b y Graphing a Plane Curve Graph the plane curve defined by the parametric equations x t t y t t t 3 2 2 2 2 ( ) ( ) = = − ≤ ≤ (1) EXAMPLE 1 Solution For each number t t , 2 2, − ≤ ≤ there corresponds a number x and a number y. For example, when t 2, = − then x 3 2 12 2 ( ) = − = and y 2 2 4. ( ) = − = − When t 0, = then x 0 = and y 0. = Set up a table listing various choices of the parameter t and the corresponding values for x and y, as shown in Table 6. Plot these points and connect them with a smooth curve, as shown in Figure 63. The arrows in Figure 63 indicate the orientation. t x t( ) y t( ) x y, ( ) 2− 12 4− 12, 4 ( ) − 1− 3 2− 3, 2 ( ) − 0 0 0 0, 0 ( ) 1 3 2 3, 2 ( ) 2 12 4 12, 4 ( ) Table 6 Figure 63 x t t y t t t 3 , 2, 2 2 2 ( ) ( ) = = − ≤ ≤ x 5 10 (0, 0) (3, 2) (12, 4) (12, 24) (3, 22) y 4 24 DEFINITION Parametric Equations and Plane Curves Suppose x x t( ) = and y y t( ) = are two functions of a third variable t, called the parameter , that are defined on the same interval I. Then the equations x x t y y t ( ) ( ) = = where t is in I, are called parametric equations , and the graph of the points defined by x y x t y t , , ( ) ( ) ( ) ( ) = is called a plane curve . Equations of the form y f x , ( ) = where f is a function, have graphs that are intersected no more than once by any vertical line. The graphs of many of the conics, and certain other more complicated graphs, do not have this characteristic. Yet each graph, like the graph of a function, is a collection of points x y , ( ) in the xy -plane; that is, each is a plane curve . In Chapter 9, we introduced polar coordinates and used polar equations to graph some interesting plane curves. In Section 10.6 , we used polar equations as a very efficient way to graph conics. In this section, we introduce another way of representing plane curves using parametric equations .
RkJQdWJsaXNoZXIy NjM5ODQ=