Whatever happened to the conic sections: Parabolas

Most high school parabolas these days are either concave up or concave down.

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.

How do you say this? A “non-functioning” parabola? Malfunctioning?

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.