## Whatever happened to the conic sections: Parabolas

I don’t know about anywhere else, but in our school district, any parabola that is not solvable by the quadratic formula is an endangered species. Specifically, I am mourning the loss of parabolas whose directrix is not parallel to the x-axis. The ones that are parallel to the x-axis are such a mundane part of the curriculum that we never discuss the existence of a focus or a directrix.

Parabolas can open up in any direction, not just up or down. And they all possess a focus and a directrix. As an example, suppose the focus of the parabola was at the origin, and the directrix was the line $-x - y = 2$, or $y= -x-2$ if you prefer slope-intercept form. This ought to lead to a single parabola as a result: $x^2 - 2xy + y^2 -4x - 4y = 4$. Parabolas like these cannot be resolved to familiar equations of the form of $y= ax^2 + bx + c$, largely due to the difficulty in separating $x$ from $y$.

I have gotten used to saying to my students that all polynomial functions of $x$ have a domain in all real numbers. This works because I am only talking about functions. But for relations like $x^2 - 2xy + y^2 -4x - 4y = 4$, both domain and range have a restriction. The parabola looks like the illustration to the right. This relation is not a function, because  1) $x$ and $y$ can’t be separated, and 2) you can pass a vertical line through more than one point on its graph – that is, one value of $x$ generates two values of $y$. Other patterns we take for granted are also broken: the vertex is not a local extrema anymore. We require implicit differentiation to obtain the local extrema: $\frac{dy}{dx} = \frac{-x+y+2}{-x+y-2} = 0$ , which boils down to $y = x-2$. From the point of intersection between this line and the parabola, we find it will have an absolute minimum at $(1, -1)$. The range, then, is $y \ge -1$.

The vertex is located at $(-0.5, -0.5)$.

The domain is, similarly, $x \ge -1$.