sketch the line segment whose parametric equations

Question

Sketch the line segment whose parametric equations are $x=2+t, v=t^2-1, t∈[0,3]$

That's what i did

$t=x-2$

$v=(x-2)^2-1$

$v=x^2-4x+3 $

$v=(x-3)(x-1)$

$x=3,1$

and that's my sketch

I am not sure if I did it right or wrong...

Answer 1

Parametric equations are described by parameters. In your case, the parameter is $t$, and you're drawing a curve on the $x-v$ axes based on equations for $x$ and $v$ in terms of $t$.

To get the curve in just $x$ and $v$, eliminate $t$, just like you did:

$$v = (x-3)(x-1)$$

You've also sketched the entire parabola correctly if you add back in the part below the $x$ axis that you erased.

Now, we want just the part that's between $t=0$ and $t=3$. It's probably easiest to look at the values of $x$ that this corresponds to. If $t=0$ then $x=2$. If $t=3$ then $x=5$. So just include the parts of the parabola with $x$ on $[2,5]$, and you're done!

Once you've done this, you can check your work by verifying that the values of $v$ are correct for these values of $t$. (It should go from $v=-1$ to $v=8$.)