Answer
This answer requires writing and explaining in statements. I have explained in the work step by step section.
Work Step by Step
For the equation of the circle:
You need to see that $x$ and $y$ are of the second degree. ( ie they are both squared). You factorise the equation to make it into the standard form $(x-a)^{2}$+$(y-b)^{2}$=$r^2$. Any circle should satisfy this relationship.
For Parabolas:
The equation should relate $x$ to $ y$, with either of them squared
Either $x$ or $y$ should be squared.
If $x$ is squared, then the parabola is either facing up or down and is of the form $y=ax^2+bx+c$. An upward facing has the value of $a$ being a positive and a downward facing has a negative value of $a$.
If $y$ is squared then the parabola is left or right opening and of the form $x=ay^2
+by+c$ . A left opening has the value of $a$ being negative and a right opening has a positive value of $a$
