SECTION 10.7 Plane Curves and Parametric Equations 737 2 Graph Parametric Equations Using a Graphing Utility Most graphing utilities have the capability of graphing parametric equations. The following steps are usually required to obtain the graph of parametric equations. Check your owner’s manual to see how yours works. Graphing Parametric Equations Using a Graphing Utility Step 1 Set the mode to PARAMETRIC. Enter x t( ) and y t . ( ) Step 2 Select the viewing window. In addition to setting X X X min, max, scl, and so on, the viewing window in parametric mode requires setting minimum and maximum values for the parameter t and an increment setting for t T ( step). Step 3 Graph. Graphing a Curve Defined by Parametric Equations Using a Graphing Utility Graph the curve defined by the parametric equations x t t y t t t 3 2 2 2 2 ( ) ( ) = = − ≤ ≤ (1) Solution EXAMPLE 2 Step 1 Enter the equations x t t y t t 3 , 2 2 ( ) ( ) = = with the graphing utility in PARAMETRIC mode. Step 2 Select the viewing window. The interval I is t 2 2, − ≤ ≤ so we select the following square viewing window: T X Y min 2 min 1 min 5 = − = − = − T X Y max 2 max 15 max 5 = = = T X Y step 0.1 scl 1 scl 1 = = = We choose Tmin 2 = − and Tmax 2 = because t 2 2. − ≤ ≤ Finally, the choice for Tstep determines the number of points the graphing utility plots. For example, with Tstep at 0.1, the graphing utility evaluates x and y at t 2, 1.9, 1.8, = − − − and so on. The smaller the Tstep, the more points the graphing utility plots. Experiment with different values of Tstep to see how the graph is affected. Step 3 Graph. Notice the direction in which the graph is drawn as it appears. This direction shows the orientation of the curve. See Figure 64(a) for the graph on a TI-84 Plus CE. Figure 64(b) shows the graph obtained using Desmos. To graph parametric equations using Desmos, the x and y equations must be entered as an ordered pair. Figure 64 x t t y t t t 3 , 2, 2 2 2 ( ) ( ) = = − ≤ ≤ 5 25 21 15 (a) (b)
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