PP 11.1 - Curves Defined by Parametric Equations Definition...

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Curves Defined by Parametric Equations
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Definition Suppose both x and y are functions of a third variable t (called a parameter ). Then for each value of t a point ( x(t), y(t)) in the plane is determined. The curve C consisting of all those points is a parametric curve . 5 0 , , 1 : Example 2 = + = t t y t x A first attempt for sketching a parametric curve can be done by plotting points. That requires a table identifying the coordinates of the points as the parameter changes.
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t x y (x, y) 0 1 0 (1, 0) 1 2 1 (2, 1) 2 3 4 (3, 4) 3 4 9 (4, 9) 4 5 16 (5, 16) 5 6 25 (6, 25) 5 0 , , 1 2 = + = t t y t x
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Plotting the points Parametric Curve 5 0 , , 1 2 = + = t t y t x
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Another possibility is to eliminate the parameter to get a curve described as before. 5 0 , , 1 2 = + = t t y t x 2 2 ) 1 ( 1 1 - = = - = + = x t y x t t x 6 1 5 0 , 1 + = x t t x 6 1 , ) 1 ( 2 - = x x y
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π 2 0 ), sin( ), cos( : = = t t y t x C Parametric equations can be used to describe curves even if they are not the graph of functions. function.
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PP 11.1 - Curves Defined by Parametric Equations Definition...

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