Hyperbolas

A hyperbola is simply a pair of curves that open away from each other, like two bowls placed back to back. Its equation is the ellipse's with a minus sign in the middle, and that one sign changes everything. The pieces are the same as before: $h$ and $k$ (the center, sitting between the branches), $a$ (how far from the center each branch starts) and $b$ (which, together with $a$, sets how steeply the branches spread). The asymptote formula is just two straight lines through the center with slope $\pm\frac ba$. Bear with it; it's a close cousin of the ellipse.

Why there's a gap in the middle. Take $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. The $y$ part is subtracted, so $\frac{x^2}{a^2}$ has to be at least $1$, meaning $|x| \ge a$. Nothing on the curve lives between $x = -a$ and $x = a$, which is why it splits into two branches starting at $(\pm a, 0)$.

Why the asymptotes are there. Far out, $x$ and $y$ are both huge, so the $1$ on the right is tiny by comparison. The equation is then roughly $\frac{x^2}{a^2} = \frac{y^2}{b^2}$, which means $y \approx \pm\frac bax$. So the branches get closer and closer to those two lines.

Which way it opens. The positive squared term wins. If the $x$ term is positive, the branches open left and right, with vertices $(h \pm a, k)$. If the $y$ term is positive, as in $\frac{(y - k)^2}{b^2} - \frac{(x - h)^2}{a^2} = 1$, they open up and down, with vertices $(h, k \pm b)$.

Sketching. Draw a box reaching $a$ across and $b$ up and down from the center, and extend its diagonals; those are the asymptotes. Then draw each branch from its vertex, bending toward them.

The LORAN navigation system located ships on hyperbolas of constant time difference; cooling towers and some telescope mirrors are hyperbolic.