Parabolas

A parabola is simply the U-shaped curve you get from graphing any quadratic, the same shape a thrown ball traces through the air. Vertex form, $a(x - h)^2 + k$, has three small pieces: $a$ (how steep the U is, and whether it opens up or down), $h$ (how far left or right the bottom of the U sits) and $k$ (how high it sits). The second formula, $h = -\frac{b}{2a}$, is just a shortcut to find that bottom point when the quadratic is written the ordinary way. Once you see those three numbers, the whole curve follows.

Why vertex form shows the vertex. Look at $a(x - h)^2 + k$ with $a > 0$. The squared part can't be negative, and it's zero exactly when $x = h$. So the smallest value is $k$, reached at $x = h$, and that's the bottom of the U. With $a < 0$ the same reasoning makes $k$ the largest value instead. And because $(x - h)^2$ is the same a step left or a step right of $h$, the parabola is symmetric about the line $x = h$.

Graphing a parabola:

1. Find the vertex: $h = -\frac{b}{2a}$, then $k = f(h)$.
2. Find the y-intercept $(0, c)$ and reflect it across $x = h$ for a free second point.
3. Find the x-intercepts, if any, by solving $f(x) = 0$.
4. Sketch: up if $a > 0$, down if $a < 0$.

Getting to vertex form. Complete the square: $x^2 + 6x + 5 = (x^2 + 6x + 9) - 9 + 5 = (x + 3)^2 - 4$, so the vertex is $(-3, -4)$. Adding and subtracting 9 doesn't change anything; it just regroups the expression to show the square.

Since the vertex is where a quadratic reaches its highest or lowest value, it's the answer to every "maximize the area" or "maximize the revenue" problem built on a quadratic.

Thrown objects, satellite dishes and car headlights are parabolas: the reflective property sends parallel rays through one focus point.