Parametric equation: $x=t^2, y=t$

Question

Why there is a curve below x-axis? I thought that because square root is involved, y should be >= 0.

Answer 1

$y=t$ can be negative. And $y<0$ is below $x$-axis. And $x=t^2 \ge 0$ so it lies right of the $y$-axis.

Answer 2

This is because this is a little different than saying $y=x^2$. What we are basically doing is expressing the coordinates in terms of one parameter $t$. Here, we have that the coordinate $(x,y)$ is expressed as $(t^2, t)$. Hence, as $t^2 \geq 0$, the $x$ coordinate can never be less than zero, however, as the y coordinate $y=t$, we have that $y$ can be negative.

Example computations: let's walk along $t$ to see how this curve evolves.

for $t = 1$, we have $(x,y) = (1^2,1) = (1,1)$

for $t = 0$, we have $(x,y) = (0^2,0) = (0,0)$

for $t = -1$, we have $(x,y) = ((-1)^2,-1) = (1,-1)$

As you can see, we have a point lying beneath the $x$-axis