852 CHAPTER 8 Applications of Trigonometry Basic Concepts We have graphed sets of ordered pairs that correspond to a function of the form y = ƒ1x2 or r = g1u2. Another way to determine a set of ordered pairs involves the equations x = ƒ1t2 and y = g1t2, where t is a real number in an interval I. Each value of t leads to a corresponding x-value and a corresponding y-value, and thus to an ordered pair 1x, y2. 8.8 Parametric Equations, Graphs, and Applications ■ Basic Concepts ■ Parametric Graphs and Their Rectangular Equivalents ■ The Cycloid ■ Applications of Parametric Equations Parametric Equations of a Plane Curve A plane curve is a set of points 1x, y2 such that x = ƒ1t2, y = g1t2, and f and g are both defined on an interval I. The equations x =ƒ1t2 and y =g1t2 are parametric equations with parameter t. Graphing calculators are capable of graphing plane curves defined by parametric equations. The calculator must be set to parametric mode. 7 Parametric Graphs and Their Rectangular Equivalents EXAMPLE 2 Finding an Equivalent Rectangular Equation Find a rectangular equation for the plane curve of Example 1, x = t2, y = 2t + 3, for t in 3-3, 34. SOLUTION To eliminate the parameter t, first solve either equation for t. y = 2 t + 3 Choose the simpler equation. 2 t = y - 3 Subtract 3 and rewrite. t = y - 3 2 Divide by 2. This equation leads to a unique solution for t. EXAMPLE 1 Graphing a Plane Curve Defined Parametrically Let x = t2 and y = 2t + 3, for t in 3-3, 34. Graph the set of ordered pairs 1x, y2. ALGEBRAIC SOLUTION Make a table of corresponding values of t, x, and y over the domain of t. Plot the points as shown in Figure 74. The graph is a portion of a parabola with horizontal axis y = 3. The arrowheads indicate the direction the curve traces as t increases. –3 3 6 9 –3 3 6 9 x y (9, 9) (9, –3) x = t2 y = 2t + 3 for t in [–3, 3] Figure 74 t x y -3 9 -3 -2 4 -1 -1 1 1 0 0 3 1 1 5 2 4 7 3 9 9 GRAPHING CALCULATOR SOLUTION We set the parameters of the TI-84 Plus as shown to obtain the graph. See Figure 75. −4 −2 10 10 Figure 75 Duplicate this graph and observe how the curve is traced. It should match Figure 74. S Now Try Exercise 9(a).
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