+---
+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 $P(1,1)$
$t$ is the parameter
+To convert to cartesian, solve like simultaneous equations
+
## Polar coordinates
$$x = r\cos\theta, \quad y = r\sin\theta$$
$$r=\cos(k\theta)$$
-where
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. $r=6\cos \theta\quad$ *(multiple by $r$)*
+---
+
+e.g. $r=6\cos \theta\quad$ *(multiply by $r$)*
+
$r^2=6r\cos\theta$
+
$x^2+y^2=6x$
+
complete the square