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**Unformatted text preview: **Parametric Equations 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 , , 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. t x y (x, y) 1 1 (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 , , 1 2 ≤ ≤ = + = t t y t x Plotting the points Parametric Curve 5 , , 1 2 ≤ ≤ = + = t t y t x Another possibility is to eliminate the parameter to get a curve described as before. 5 , , 1 2 ≤ ≤ = + = t t y t x 2 2 ) 1 ( 1 1- = = ⇒- = ⇒ + = x t y x t t x 6 1 5 , 1 ≤ ≤ ⇒ ≤ ≤ + = x t t x 6 1 , ) 1 ( 2 ≤ ≤- = ⇒ x x y Comments π 2 ), 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. a represent NOT does curve is Clearly th circle. unit the is which 1 have we , 1 cos sin Since 2 2 2 2 , y x (t) (t) = + = + Parametric description of curves is particularly useful when describing motion or particles.useful when describing motion or particles....

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