Parabola
A parabola is the U-shaped graph of a quadratic function, characterized by a vertex, axis of symmetry, and opening either upward or downward.
Formula
y = ax^2 + bx + c
Definition
A parabola is the U-shaped curve produced by graphing a quadratic function $y = ax^2 + bx + c$: it opens upward (a smile, with a minimum) when $a > 0$ and downward (a frown, with a maximum) when $a < 0$, with key features including the vertex, axis of symmetry, y-intercept $(0, c)$, and x-intercepts (the real roots, if any exist). More formally, a parabola is a conic section, the locus of points equidistant from a fixed point (the focus $F$) and a fixed line (the directrix $d$); for $y = \frac{1}{4p}x^2$, the focus is at $(0, p)$ and the directrix is $y = -p$, its eccentricity is exactly $1$, and it has the reflective property that rays parallel to the axis all pass through the focus after reflection.
Example
$y = x^2$ opens upward; $y = -x^2$ opens downward, and the tossed path of a ball traces a parabola. For $y = x^2 - 4$, the vertex is $(0, -4)$, the axis is $x = 0$, and the x-intercepts are at $x = 2$ and $x = -2$. Writing $y = x^2$ as $y = \frac{1}{4p}x^2$ gives $4p = 1$, so $p = 1/4$: focus at $(0, 1/4)$, directrix $y = -1/4$.
Key Insight
Parabolas are everywhere: satellite dishes, the path of a thrown ball, and the cross-section of car headlight reflectors are all parabolic. The number of x-intercepts reveals the discriminant's sign, two means $D > 0$, one means $D = 0$ (vertex touching the x-axis), zero means $D < 0$. That reflective property is exactly why satellite dishes and headlights are shaped like parabolas: incoming parallel signals (or outgoing light) all focus at, or emerge from, that single focus point.