Is $\gamma(t) = (|t|,t)$ plot $y = x$?

Question

I have this parametric curve :

$\gamma(t) = (|t|,t)$ with $\gamma(t) : \mathbb{R} \to \mathbb{R}^2$

And I have to say if the plot is the line of equation $y = x$. Here's my answer:

$x(t) = |t|$ and $y(t) = t$

if $y = x$, $y(t) = x(t)$ on $\mathbb{R}$

or $t = |t|$ if and only if $t \ge 0$

So it's false. Is my answer correct?

Answer 1

Yes, you're correct. For $t \geq 0$, $\gamma(t) = (t,t)$ is indeed the line $y = x$. But for $t < 0$, $\gamma(t) = (-t,t)$. It is the graph of $f(x) = |x|$ rotated around the origin $90$ degrees clockwise.