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plot(x,y)
axis square

The parametric representation of many common curves is our next topic of
interest. The parametric representation is defined such that if x and y are con-
tinuous functions of t over the interval I, we can describe a curve in the x-y
plane by specifying:

C: x = x(t), y = y(t), and t ∈ I

More Examples:
In the following examples, we want to identify the curves f(x, y) = 0 corre-
sponding to each of the given parametrizations.

Example 1.12
C: x = 2t – 1, y = t + 1, and 0 < t < 2. The initial point is at x = –1, y = 1, and the
final point is at x = 3, y = 3.

Solution: The curve f(x, y) = 0 form can be obtained by noting that:

2t – 1 = x ⇒ t = (x + 1)/2

Substitution into the expression for y results in:

y = x + 3
2 2

This describes a line with slope 1/2 crossing the x-axis at x = –3.

Question: Where does this line cross the y-axis?

Example 1.13
C: x = 3 + 3 cos(t), y = 2 + 2 sin(t), and 0 < t < 2π. The initial point is at x = 6, y
= 2, and the final point is at x = 6, y = 2.

Solution: The curve f(x, y) = 0 can be obtained by noting that:

y − 2 x − 3
sin( )t = and cos t () =
2 3
2
2
Using the trigonometric identity cos (t) + sin (t) = 1, we deduce the following
equation:

© 2001 by CRC Press LLC