A parabola is the locus of points $P$ in the Euclidean plane which are equidistant from a given line $\ell$ , called the directrix, and a given point $F$ not on the directrix, called the focus.

To obtain a simple equation for the parabola, assume that the directrix is parallel to the $x$ -axis, the focus is on the $y$ -axis, and the directrix and focus are the same distance from the origin. By reflecting the plane if necessary, this means that there is a positive number $a$ such that the equation of the directrix $\ell$ is $y = -a$ and the position of the focus $F$ is $(0, a)$ . Then the condition that a point $(x,y)$ is equidistant from $\ell$ and $F$ can be interpreted as the equation $$ (y + a)^2 = x^2 + (y - a)^2. $$ Since $(y + a)^2 - (y - a)^2 = 4ay$ , the above equation simplifies to $$ y = \frac{1}{4a}x^2. $$

Below is the graph of a parabola for $a=1$ :

From the equation $$ y = \frac{1}{4a}x^2, $$ we can immediately observe some important properties of the parabola. First, since $x^2$ is an even function, the parabola is symmetric with respect to the $y$ -axis; this can also be deduced directly from the geometric definition of the parabola. The intersection point of the parabola and the symmetry axis, is called the apex of the parabola; it is the point of the parabola nearest the directrix. Second, notice that the coefficient of $x^2$ in the equation of the parabola is inversely proportional to $2a$ , the distance between the focus and the directrix. So this distance controls how rapidly the function $\frac{1}{4a}x^2$ grows. As $a$ tends to zero, the parabola becomes flatter and flatter, tending to the straight line $y = 0$ in the degenerate case $a = 0$ . On the other hand, as $a$ increases, the curvature of the parabola at $0$ increases. When $a$ tends to infinity, the parabola begins to resemble a hairpin more and more until it suddenly becomes a single point, the origin, in the degenerate case $a = \infty$ .

The parabola is a conic section with eccentricity 1. All parabolas are similar, which follows directly from the definition of parabola.