Is there a parabola passing through points (00), (1,0), and (0,1)? Answer the same for an ellipse and for a hyperbola. 
This is the work I've done so far. I plugged in the x and y values into each conic's standard form equation. But I'm not sure where to go from here. I've been really confused with conics.
 A: Construct a family of conics with intercepts $(0,0)$, $(1,0)$ and $(0,1)$:
$$ax^2+2hxy+by^2-ax-by=0$$
$$\Delta=\det
\begin{pmatrix}
  a & h & -\frac{a}{2} \\
  h & b & -\frac{b}{2} \\
  -\frac{a}{2} & -\frac{b}{2} & 0 \\
\end{pmatrix}
=\frac{ab(2h-a-b)}{4} \\$$

*

*$ab-h^2>0 \implies$ ellipse (always real)


*$ab-h^2=0$ and $\Delta\ne 0 \implies$ parabola


*$ab-h^2<0$ and $\Delta\ne 0 \implies$ hyperbola


*$a=0 \implies y(2hx+by-b)=0$, two straight lines


*$b=0 \implies x(ax+2hy-a)=0$, two straight lines


*$h=\frac{a+b}{2} \implies (x+y-1)(ax+by)=0$, two straight lines
See another post of mine for your interests.
A: General conic: $g(x,y)=Ax^2+Bx+Cy^2+Dy+Exy+F=0$
$g(0,0)=0\implies F=0.$
$g(0,1)=0 \implies C+D=0\implies D=-C$
$g(1,0)=0 \implies A+B=0 \implies B=-A$
So: $g(x,y)=A(x-1/2)^2+C(y-1/2)^2+Exy -(A+C)/4=0$
Any choice of A,C, and E will give you a conic with the three mentioned points on it. Let $E=0$ and $A=C=1$, we have a selection of coefficients that produces a circle.
More generally, using the quadratic formula to solve for $y$ in terms of $x$:
$$y=\frac{(C-Ex)+\sqrt{(E^2-4AC)x^2+(4AC-2EC)x+C^2}}{2C}$$
Here the discriminant is a function of $x$ that represents a parabola with axis parallel to the y axis.
Given a parabola of the form $y=mx^2+nx+p$, it  has a minimum if m is positive, a max if m is negative. In either case, the extremal value of y is $y_e=-n^2/4m+p$.
Properties of this parabola tell you about the conic. If it's a perfect square, the conic is a line. If its negative sufficiently far from the origin, the conic is an ellipse. Otherwise its a hyperbola or a parabola, largely depending on if it touches or crosses the x axis. Parabola if touching, hyperbola if  its positive for all points above a certain distance from the origin.
