Focus and directrix of a parabola

Find the focus and directrix of the parabola $y = \dfrac{x^2}{4}$ ¹. Rewriting it as $x^2 = 4y$ matches the form $x^2 = 4py$ with $p = 1$ , so the focus is $(0, 1)$ and the directrix is the line $y = -1$ .

Equidistance

Every point on the parabola is equidistant from the focus and the directrix.

Point on the curve	Distance to the focus	Distance to the directrix
$(0, 0)$	$1$	$1$
$(2, 1)$	$2$	$2$
$(4, 4)$	$5$	$5$

The definition gives the equation

That equidistance is in fact the definition of a parabola. If a point $(x, y)$ is equidistant from the focus and the directrix, then the following holds.

\sqrt{x^2 + (y - 1)^2} = |y + 1|

Squaring both sides gives $x^2 + y^2 - 2y + 1 = y^2 + 2y + 1$ ; the $y^2$ and the $1$ cancel, leaving $x^2 = 4y$ . The equation of a parabola is nothing but this equidistance rewritten. Just as a circle is defined by equal distance from a center, a parabola is defined by equal distance from a point and a line.

How wide the parabola opens

The points $(2, 1)$ and $(-2, 1)$ are the ends of the chord through the focus perpendicular to the axis. Its length is $4p = 4$ , and it measures how widely the parabola opens: the larger $p$ , the farther the focus and the broader the curve.

Parabola	$p$	Focus	Directrix
$y = x^2$	$\dfrac{1}{4}$	$\left(0, \dfrac{1}{4}\right)$	$y = -\dfrac{1}{4}$
$y = \dfrac{x^2}{4}$	$1$	$(0, 1)$	$y = -1$

Why it is called the focus

The name comes from optics. Every ray arriving parallel to the axis is reflected by the parabola through that single point².

Satellite dishes and reflecting telescopes gather parallel rays into the focus
Run backwards, a source at the focus sends out rays parallel to the axis
That is how the reflector behind a headlight or a flashlight bulb works

The large dots mark the focus and the two ends of the chord through it.

Parabola, Wikipedia
Parabolic reflector, Wikipedia