SECTION 10.7 Plane Curves and Parametric Equations 743 5 Find Parametric Equations for Plane Curves Defined by Rectangular Equations If a plane curve is defined by the function y f x , ( ) = one way of finding parametric equations is to let x t. = Then y f t , ( ) = and x t t y t f t t f in the domain of ( ) ( ) ( ) = = are parametric equations of the plane curve. Finding Parametric Equations for a Plane Curve Defined by a Rectangular Equation Find two different pairs of parametric equations for the function y x 4. 2 = − Solution EXAMPLE 7 For the first pair of parametric equations, let x t. = Then the parametric equations are x t t y t t t 4 2 ( ) ( ) = = − −∞< <∞ A second pair of parametric equations is found by letting x t .3 = Then the parametric equations become x t t y t t t 4 3 6 ( ) ( ) = = − −∞< <∞ Care must be taken when using the second approach in Example 7. The substitution for x must be a function that allows x to take on all the values in the domain of f. For example, letting x t t 2 ( ) = so that y t t 4 4 ( ) = − does not result in equivalent parametric equations for y x 4, 2 = − since only points for which x 0 ≥ are obtained; yet the domain of y x 4 2 = − is x x is any real number . { } Now Work PROBLEM 33 Solution Finding Parametric Equations for an Object in Motion Find parametric equations for the ellipse x y 9 1 2 2 + = where the parameter t is time (in seconds) and (a) The motion around the ellipse is clockwise, begins at the point 0, 3 , ( ) and requires 1 second for a complete revolution. (b) The motion around the ellipse is counterclockwise, begins at the point 1, 0 , ( ) and requires 2 seconds for a complete revolution. EXAMPLE 8 Figure 74 x y 9 1 2 2 + = with clockwise orientation y x (0, 23) (1, 0) (21, 0) (0, 3) (a) Figure 74 shows the graph of the ellipse. Since the motion begins at the point 0, 3 , ( ) we want x 0 = and y 3 = when t 0. = Let x t t y t t sin and 3cos ω ω ( ) ( ) ( ) ( ) = = for some constant .ω These parametric equations satisfy the equation of the ellipse. They also satisfy the requirement that when t 0, = then x 0 = and y 3. = For the motion to be clockwise, the motion has to begin with the value of x increasing and the value of y decreasing as t increases. This requires that 0. ω > [Do you know why? If 0, ω > then x t t sin ω ( ) ( ) = is increasing when t 0 ≥ is near zero, and y t t 3cos ω ( ) ( ) = is decreasing when t 0 ≥ is near zero.] See the red part of the graph in Figure 74. Finally, since 1 revolution takes 1 second, the period 2 1, π ω = so 2 . ω π = Parametric equations that satisfy the conditions stipulated are x t t y t t t sin 2 3cos 2 0 1 π π ( ) ( ) ( ) ( ) = = ≤ ≤ (3) Need to Review? The period of a sinusoidal graph is discussed in Section 6.4, pp. 430–432. (continued)
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