Finding at least $4$ parameterizations of the curve $y=x^2-6x+9$. I can only find two.

Question

Finding at least $4$ parameterizations of the curve defined by $y=x^2-6x+9$

What I tried:

• One way: Let $x=t,$ then $y=(t-3)^2$. So coordinates of any point on the curves is $(t,(t-3)^2)$.

• Another way: Let $x=t+3,$ then $y=t^2$. So any point on the curves is $(t+3,t^2)$.

But I do not understand how I can parameterize in two other different ways.

Help me please, Thanks.

Answer 1

You can set $$ x=\alpha t+\beta, \quad \alpha\neq0 $$ so that $$ y=(\alpha t+\beta)^2-6(\alpha t+\beta)+9 $$ and you have how many parametrizations as you want, choosing different values of $\alpha,\beta.$

Your parametrizations correspond to $\alpha=1,\beta=0,$ and to $\alpha=1,\beta=3.$

Instead of a linear function of $t$ you can choose any bijective function of $I\to\mathbb{R},$ where $I$ is an interval.