\[y=f(x)\]
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. \[(x,y)=(t^2e^{-t}-t, 2+\frac{1+e^t}{2-t} )\]
.