Parametric Equations

We can define a curve
$y=f(x)$
para metrically by introducing a convenient parameter which defines a point on a curve. The parameter may define for example the length along the curve from a given point, and this is useful for some purposes.
Example: We can define a circle radius
$5$
centred at the point
$(2,3)$
in terns of the parameter
$\theta$
, the angle between the horizontal and a radius from the centre of the circle at
$(2,3)$
to a point
$(x,y)$
on the circle
$x=2+5 cos \theta , \; y=3+5 sin \theta$
. In this example the parameter
$\theta$
is not the length from a given point.
We can write the line


$y=2x+3$
para metrically as
$(\frac{t}{ \sqrt{5}} , \frac{2}{ \sqrt{2}} +3 )$
. In this example as
$t$
increases by 1, we move
$\sqrt{(\frac{1}{\sqrt{5}})^2+ ( \frac{2}{\sqrt{5}})^2}=1$
which is the distance along the curve.
Parametric equations are useful for more complex curves, where it could be hard to give the equation in the form
$f(x)=g(y)$
e.g.
$(x,y)=(t^2e^{-t}-t, 2+\frac{1+e^t}{2-t} )$
.