As $\quad f(x) \rightarrow \pm \infty,\quad {1 \over f(x)} \rightarrow 0^\pm$ (vert asymptote at $f(x)=0$)
-As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$
+<!-- As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$ -->
- reciprocal functions are always on the same side of $x=0$
- if $y=f(x)$ has a local max|min at $x=1$, then $y={1 \over f(x)}$ has a local max|min at $x=a$
Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
-Asymptotes at $y-k=\pm {b \over a}(x-h$)
+Asymptotes at $y=\pm {b \over a}(x-h)+k$
+To make hyperbola up/down rather than left/right, swap $x$ and $y$
+
+$y^2-x^2=1$ produces hyperbola shifted 90 $^\circ$ (top and bottom of asymptotes)
## Parametric equations
$$x=f(t), \quad y=g(t)$$
$t$ is the parameter
+
+## Polar coordinates
+
+$$x = r\cos\theta, \quad y = r\sin\theta$$
+
+### Spirals
+$$r={\theta \over n\pi}$$
+- solve intercepts for multiples of $\pi \over 2$
+- or draw table of values for $r$ and $\theta$ for each $n\pi \over 2$
+
+### Circles
+$$r=a$$
+
+### Lines
+
+Horizontal: $r={n \over \sin \theta}$
+Vertical: $r={n \over \cos \theta}$
+
+### Solving polar graphs
+
+solve in terms of $r$
+
+e.g. $x=4$
+$r\cos\theta = 4$
+$r={4 \over \cos\theta}$
+
+e.g. $y=x^2$
+$r\sin\theta = r^2 \cos^2 \theta$
+$\sin \theta = r \cos^2 \theta$
+$r = {\sin \theta \over \cos^2\theta} = \tan\theta \sec\theta$
+
+e.g. $r=6\cos \theta\quad$ *(multiple by $r$)*
+$r^2=6r\cos\theta$
+$x^2+y^2=6x$
+complete the square
+
+## Other graphs
+
+### Cardioids
+
+$$
+
+### Roses