ONTO and 1-1 for F(x,y) For the function $f: {\rm I\!R}^2 \rightarrow {\rm I\!R} $ where $F(x,y)$ = $ax^2 + bxy + cy^2 +dx +ey$ it's unclear to me how to determine which parameters ($a,b,c,d,e$) for $f(.)$ are onto and/or 1-1. For a function to be onto, there needs to be an $x$ in the domain for every $y$ in the co-domain. I think I understand this for a function $f(x) = y$, but how then do I apply this to a function $f(x,y) = z$?
It's my understanding that tor a function to be 1-1, it needs to pass the horizontal line test. I've tried graphing the function and playing with different parameter different values to see if it makes visual sense. If I give parameters $a,b$ or $c$ values, the surface takes on a saddle like shape where points on the $x$ axis intersect with more than one point on the $z$ axis. 
whereas, if I leave them zero and give parameters $d$ or $e$ values, the surface becomes flat, and the horizontal line test seems to pass. 
Flat Surface
Curved Surface
This makes sense to me visually, but I would like to know how to determine which parameters are injective and/or surjective more formally?
 A: $z = f(x,y) = ax^2 + bxy + cy^2 + dx + ey$
The function is surjective (onto) if for every $z$ in $\mathbb R$ there is some $(x,y)$ pair that maps to z. 
Clearly we can find values of $f(x,y)$ such that $z \le 0$  e.g. $a,c > 0 , b,d,e = 0$ 
For what values could it be onto?  Look at $b^2 - 4ac.$ What is this expression in the quadratic formula, and why might that be relevant here?
The function is injective (into, 1-1) if $f(x_1,y_1) = f(x_2,y_2) \implies (x_1,y_1) = (x_2,y_2).$
Fix some value of $z$ is the set of $(x,y)$ such that $f(x,y) = z $ a curve (or a line) or a singular point?
If it a curve, it can't be injective.
Update:
Suppose we set $z$ equal to some constant $k$
$a,b,c = 0 \implies dx + ey = k$ is a line.
Otherwise:
$b^2 - 4ac > 0 \implies ax^2 + bxy + cy^2 + dx + ey = k$ is a hyperbola.
$b^2 - 4ac = 0 \implies ax^2 + bxy + cy^2 + dx + ey = k$ is a parabola.
and $b^2 - 4ac < 0 \implies ax^2 + bxy + cy^2 + dx + ey = k$ is an ellipse.
In all cases $f(x,y)$ cannot be 1-1.
What about onto?
If $f(x,y) = k$ describes a sequence of hyperbola, will it be defined for all values of $k$?
What about if it a sequence of ellipses or a parabola?
Another way to think about it... 
If $b^2 - 4ac > 0$ then $(ax^2 - bxy + cy^2)$ can be factored into something of the form $(px + qy)(rx + sy).$
If $b^2 - 4ac = 0$ then $(ax^2 - bxy + cy^2)$ is a perfect square i.e. $(\sqrt a x + \sqrt cy)^2.$
If $b^2 - 4ac < 0$ then $(ax^2 - bxy + cy^2)$ can be written as the sum of two squares -- something of the form $(px + qy)^2 + (rx + sy)^2.$
