If $x=a$ is not a function, nor an equation—as in its graph is not the set of all points which satisfy that equation—, then what is it?

Question

In precalculus, I was introduced to conic sections and their equations. I learned how parabolas, for example, aren't always formed from quadratic functions and how they're oftentimes better described with the use of equations. A parabola is then better defined as the set of all points which satisfy either of the two following equations:$$(y-k)^2 = 4p(x-h)$$ $$or$$ $$(x-h)^2 = 4p(y-k)$$ That being the case, what exactly is $x=a$? It's not a function because it's not 'one-to-one' and it's not the set of all points which satisfy that equation because if that were the case then its graph would be a single coordinate-point, $(x, a)$. If it's not a function, nor an equation, then what is it? Why is $x=a$ graphed as a vertical line, and likewise, why is $y=b$ graphed as a horizontal line?

Answer 1

In fact $x=a$ is very much an equation, just as much as $y=a$. The graph is a vertical line. You can write $x=a$ as $0×y + x = a$ if you prefer to see both variables participate.

Answer 2

$x = a$ is certainly an equation, because it's of the form (left side) = (right side) where the left and right sides are expressions. Assuming $a$ represents a real constant, its graph is the set of all $(x,y)$ where $x = a$, and that is indeed a vertical straight line.

Answer 3

Everything depends on the context. In the context of plane analytic geometry, the context is sets of points which are each defined by an $(x, y)$ coordinate pair. Thus, the equation $x = a$ refers to the set of all points that satisfy that equation. That is, the line it represents is the set $\{(x,y) : x = a\} $ of all points $(x,y)$ where $x=a$ and $y$ is real but otherwise unrestricted.