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Remark 8
Sometimes, new letters are introduced to denote the two functions
of the parameter t. Like, (x, y)=(f(t),g(t)) which means x = f(t)
and y = g(t), two given functions of t. To simplify, one just stays
with x(t),y(t).
Example 17
Sketch and identify the curve defined by the parametric equations
x = t
2
− 2t, y = t +1.
Each value of t gives a point on the curve (it is advised to arrange
for the respective table of inputs and outcomes). For instance, if
t =0, then x =0,y =1and so the corresponding point is (0, 1). We
plot the points (x, y) determined by several values of the parameter
and we join them to produce a curve.
Example 18
You are advised to plot the circle centered at the origin in order
to convince yourself that parametric equations
x = R cos t, y = R sin t;0 t 2π
define such a circle (with radius R).
The parameter t is the angle (in radians) between the positive
x−semiaxis and the vector OM,whereM =(x, y). One can convert
the above parametric equations into a general equation of a circle by
squaring parametric expressions and adding them:
x
2
+ y
2
= R
2
(cos
2
t +sin
2
t) or x
2
+ y
2
= R
2
.
Example 19
Let an ellipse be given by its standard equation
x
2
a
2
+
y
2
b
2
=1.
Introduce x = a cos t, y = b sin t; 0 t 2π. These are the ellipse
parametric equations.
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