Parabola
AlgebraA 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 you get when you graph a quadratic equation like $y = x^2$. It can open upward (a smile shape) or downward (a frown shape).
Example
$y = x^2$ opens upward. $y = -x^2$ opens downward. The tossed path of a ball forms a parabola.
Key Insight
Parabolas are everywhere: the shape of satellite dishes, the path of a thrown ball, and the cross-section of car headlight reflectors all form parabolas.
Definition
A parabola is the graph of a quadratic function $y = ax^2 + bx + c$. It opens upward if $a > 0$ (has a minimum) and downward if $a < 0$ (has a maximum). Key features: vertex (turning point), axis of symmetry, y-intercept $(0, c)$, and x-intercepts (the real roots, if they exist).
Example
$y = x^2 - 4$: vertex $(0, -4)$, axis $x = 0$, x-intercepts at $x = 2$ and $x = -2$ (since $x^2 = 4$). The parabola crosses the x-axis at two points.
Key Insight
The number of x-intercepts of a parabola tells you the discriminant's sign: two x-intercepts means $D > 0$, one means $D = 0$ (vertex on x-axis), zero means $D < 0$ (never touches x-axis).
Definition
A parabola is a conic section defined geometrically as the locus of points equidistant from a fixed point (focus $F$) and a fixed line (directrix $d$). For the parabola $y = \frac{1}{4p}x^2$, the focus is at $(0, p)$ and the directrix is $y = -p$. The eccentricity of a parabola is exactly $1$. Parabolas are invariant under the group of parabolic symmetries and have the reflective property: rays parallel to the axis all pass through the focus after reflection.
Example
Parabola $y = x^2$: in standard form $y = \frac{1}{4p}x^2$ with $4p = 1$, so $p = 1/4$. Focus at $(0, 1/4)$, directrix $y = -1/4$.
Key Insight
The reflective property of parabolas is the reason satellite dishes and car headlights use parabolic shapes: all incoming parallel signals (or outgoing light rays) are focused at (or emitted from) the single focus point.