College Trigonometry, 2011a

11.10 Parametric Equations 1053
Now that we have had some good practice sketching the graphs of parametric equations, we turn
to the problem of finding parametric representations of curves. We start with the following.
Parametrizations of Common Curves
ˆ To parametrize y = f(x) asx runs through some interval I, letx = t and y = f(t) andlet
t run through I.
ˆ To parametrize x = g(y) asy runs through some interval I, letx = g(t) andy = t and let
t run through I.
ˆ To parametrize a directed line segment with initial point (x 0 ,y 0 ) and terminal point (x 1 ,y 1 ),
let x = x 0 +(x 1 − x 0 )t and y = y 0 +(y 1 − y 0 )t for 0 ≤ t ≤ 1.
ˆ To parametrize (x−h)2 + (y−k)2 =1wherea, b > 0, let x = h + a cos(t) andy = k + b sin(t)
a 2 b 2
for 0 ≤ t