I have two variables, x and y, that I would like to make a 3D equation/graph out of. Both variables are a part of their own parabolic equation: $z=-\frac{1}{4}x^{2}+\frac{7}{4}x$ and $z=-\frac{1}{4}y^{2}+\frac{7}{4}y$. I don't want the intersection of the two equations in 3D space: I want the two equations to interact with and affect each other.
Here is a picture of the current graph: Two parabolas intersect in 3D space. The single 3D function I have in mind would look more like a ball under a blanket or a 3D bell curve, with the vertex in the exact same place as the vertex of the two existing parabolas (coordinates $(3.5, 3.5, 3.063)$), and wherever either or both $x$ and $y$ are $0$, $z$ is $0$.
As an example, here are the list of coordinates I would like this equation to produce:
- $(3.5, 3.5, 3.063)$ - the vertex
- $(0, 0, 0)$
- $(7, 0, 0)$
- $(0, 7, 0)$
- $(7, 7, 0)$
How do I solve for this equation? I understand how to solve for a parabola from a set of coordinates, but I do not know how to deal with polynomials with more than one variable, as I have never taken a multivar class. Thanks in advance!
Considering
$$ F(x,y,z) = \frac 14(x^2+y^2)-\frac 74(x+y)+2z=0 $$
we have the graphics. In black the points
$$ \{(3.5, 3.5, 3.063),(0, 0, 0),(7, 0, 0),(0, 7, 0),(7, 7, 0)\} $$