# How do I eliminate the parameter to find a Cartesian equation?

Given the two parametric equations

$$x=t^2+1$$

$$y=t+1 \iff t= y-1$$

How do I eliminate parameter $$t$$ to find a Cartesian equation? Do I substitute?

This is confusing me, so I would appreciate it if somebody could explain how to do this. Thank you for your time.

• You can get $t$ from $s$ also. Then eliminate $t$ from the two relations. – Narasimham Dec 10 '18 at 21:59

Both $x$ and $y$ are functions of $t$.
Solving $y = t+1$ to obtain $t$ as a function of $y$: we have $t = y-1.\quad$
So given $x=t^2 + 1$, by substitution of $t = (y-1)$, we have $$x=(y-1)^2 +1 \iff x-1=(y-1)^2$$
We have a horizontal parabola with vertex at $(1, 1)$ and opening to the right (positive direction.)