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Section 0.4: Parametric Curves

We have discussed several diﬀerent types of functions. What do we do with those curves that

are not functions? It turns out that we can describe such curves using a pair of functions,

one for each variable.

Def: Let f(t)and g(t)be two functions deﬁned on the interval t∈[a, b]. Then the curve in

the xy-plane given by

(x, y) = (f(t), g(t)) , t ∈[a, b]

is called a parametric curve, and tis the parameter along the curve.

Given a parametric curve with x=f(t)and y=g(t), we can solve both equations for tand

set them equal to each other or substitute one into the other. This will give us the usual

Cartesian equation in only xand y.

Ex: Let x=t+ 1 and y=t2. Find the Cartesian equation (eliminate the parameter t).

If x=t+ 1, then t=x−1. Then

y= (x−1)2.

This is a vertical parabola centered at (1,0).

Ex: Graph the parametric curve given by x= 4 cos tand y= 3 sin tfor t∈[0,2π). Then

ﬁnd the Cartesian equation.

To graph this, use a table and choose a selection of tvalues:

t x y

0 4 0

π/20 3

π-4 0

3π/20 -3

≤ <

1

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