+---
+geometry: margin=2cm
+columns: 2
+graphics: yes
+---
+
# Graphing techniques
## Reciprocal continuous functions
<!-- As $\quad x \rightarrow \pm \infty,\quad {-1 \over x}$ -->
+\includegraphics[width=0.25\textwidth]{./graphics/recip-parabola.png}
+\includegraphics[width=0.25\textwidth]{./graphics/recip-sin-cos.png}
+
- 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 $x=a$
- point of inflection at $P(1,1)$
Cartesian equation for hyperbolas ($a$ and $b$ are dilation factors):
$${(x-h)^2 \over a^2} - {(y-k)^2 \over b^2} = 1$$
+Distance between vertices is $2a$
+Vertices given by $(h \pm a, k)$
+
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
Parametric curve:
$$x=f(t), \quad y=g(t)$$
$t$ is the parameter
+
+To convert to cartesian, solve like simultaneous equations
+
+## 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: $r={n \over \cos \theta}$
+
+### Cardioids
+
+$$r=a(n+ \cos\theta)$$
+
+### Roses
+
+$$r=\cos(k\theta)$$
+
+If $k$ is odd, half of the petals will overlap (hence there are $n$ petals)
+
+If $k$ is even, petals will not overlap (hence $2n$ petals)
+
+\includegraphics[width=0.5\textwidth]{./graphics/rose.png}
+
+
+### 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. $r=6\cos \theta\quad$ *(multiply by $r$)*
+
+$r^2=6r\cos\theta$
+
+$x^2+y^2=6x$
+
+complete the square