Plotting for solution for $y=x^2$ and $x^2 + y^2 = a $ Consider the system $$y=x^2$$ and $$x^2 + y^2 = a $$for $x>0$, $y>0$, $a>0$. 
Solving for equations give me $y+y^2 = a$, and ultimately $$y = \frac {-1 + \sqrt {4a+1}} {2} $$ (rejected $\frac {-1 - \sqrt {4a+1}} {2} $ since $y>0$).
The next part is to plot on the $x-y$ plane for different values of $a$. Is plotting the graph of $y = x^2$ insufficient?

Yes, it is insufficient.
You should notice that this equation is "special:"
$$x^2 + y^2 = a$$
This is the graph of a circle, radius $\sqrt{a}$.
So, your graph should contain both the parabola and the part of the circle in the region in question.
Here's a link to a graph from Wolfram Alpha which may help give some intuition.  The darkest shaded region that is there is the region of interest. 
