When I was taking Geometry and learning about parabolas, I asked my teacher what the equation was for diagonal parabolas. To my disappointment, he did not give me a satisfying answer. He told me that I would learn how to find those in college, but for now, all I needed to know were the equations for parabolas that opened exclusively vertically or horizontally. I was somewhat upset when I derived the equation for a diagonal parabola using algebra I learned my sophomore year in high school.

A parabola is the set of points that are equidistant from a separate point, called a focus, and a line, called a directrix.

If I want the parabola to open up or down, I make the directrix a horizontal line. If I want the parabola to open sideways, I make the directrix a vertical line. If I made the directrix a diagonal line, then I could find a diagonal parabola by utilizing the distance formula.

The definition of a parabola is that at any one point, it is equidistant from a point and a line. I will make a point on the parabola (x,y) and set the distances from (x,y) to the directrix and to the focus equal to each other.

The left side of the equation is just the distance between two points. The right side is the formula for the distance between a point and a line. I pinky-promise I will derive that formula later, but I’m not going to do it here. I will make the equation of the line x + y = 0 so the calculations aren’t excessively painful. I’m not going to find the generic equation for a diagonal parabola; I just want one diagonal parabola.

At the end, I get an equation that’s different from the parabola formula I’m used to in geometry. This is mostly because I cannot get this equation to express y as a function of x or vice versa. I can show a vertical parabola as a function of x because for every x value, there is only one y value. That’s the definition of a function. Diagonal parabolas cannot be functions because there are always two y values for each x value (except for one point at the beginning of the parabola).

That ticks me off because I spent twenty minutes trying to find a functional diagonal parabola before it hit me. I wasted half my study hall for nothing.

I wouldn’t have done any of this if my Geometry teacher had just said that diagonal parabolas would take some more complicated algebra. Instead, I felt the need to figure it out before I went away to college. CDR, if you ever read this, I want you to know that the whole point of this post was to prove you wrong.

Update: this can be written as two functions. I can use the quadratic formula to solve for y. Thank you, Potter.

https://amsi.org.au/ESA_Senior_Years/SeniorTopic2/2a/2a_2content_10.html