So there's this question:
Write a parameterized curve whose C is part of the equation $x=y^2-1$ between the points $A=(0,-1)$ and $B=(0,1)$.
I've always had some difficulty in parametrizing some equations, but usually ones like this I can easily solve. For which I think is $\gamma(t)=(t^2-1,t)$, for example.
I'm just not sure what I have to do with the points A and B in this case and there is an obs. at the end that states "The solution is not unique".
Can someone help me out?
Given a curve between two points, there may be parameters that work and parameters that don't. Some parametrisations won't be well-defined, because for a given value of parameter, we want to get two values.
For example, if I'd tried to parametrise by $x$ in your example (instead of parametrising by $y$ as you did), that wouldn't have worked. If you weren't thinking too carefully you might write $$\gamma(t) = (t, \sqrt{t+1})$$ You can tell this is wrong by looking at the curve $C$: for some values of $x$ there are two values of $y$, whereas this parametrisation only hits half of $C$. Note that for any $A, B$ both with $y>0$, this parametrisation will work (for suitable $t$).
Another point here about $A$ and $B$, which will help you avoid this error, is that you should specify the domain for $\gamma$. In your example, $\gamma : [-1,1] \rightarrow C$.
Parametrisation isn't unique because you could also use e.g. $y-a$ for any constant $a$, or many other choices.
