[methods] logarithm laws

diff --git a/methods/transformations.md b/methods/transformations.md
index 05f5c3b08cd964cdd7d2f1c49afdc993b61b07dc..08ba3123d6c2e6fa91385082229dc569ff2b094a 100644 (file)
--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -1,3 +1,8 @@
+---
+geometry: margin=2cm
+author: Andrew Lorimer
+---
+
 # Transformation
 
 **Order of operations:** DRT - Dilations, Reflections, Translations
@@ -9,6 +14,7 @@
 - $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
 - $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
 - for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
+- when $0 < |a| < 1$, graph becomes closer to axis
 
 ## Translations
 
@@ -54,19 +60,19 @@ Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/parabola.png){#id .class width=50%} | ![](graphics/cubic.png){#id .class width=50%} |
+|![](graphics/parabola.png){#id .class width=20%} | ![](graphics/cubic.png){#id .class width=20%} |
 
 ### $x^n$ where $n \in \mathbb{Z}^-$
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/truncus.png){#id .class width=50%} | ![](graphics/hyperbola.png){#id .class width=50%} |
+|![](graphics/truncus.png){#id .class width=20%} | ![](graphics/hyperbola.png){#id .class width=20%} |
 
 ### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
 
 | $n$ is even: | $n$ is odd: |
 | ------------ | ----------- |
-|![](graphics/square-root-graph.png){#id .class width=50%} | ![](graphics/cube-root-graph.png){#id .class width=50%} |
+|![](graphics/square-root-graph.png){#id .class width=20%} | ![](graphics/cube-root-graph.png){#id .class width=20%} |
 
 
 ### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
@@ -82,8 +88,8 @@ If $n$ is odd, it is an odd function.
 
 $$x^{p \over q} = \sqrt[q]{x^p}$$
 
-- if $p \gt q$, the shape of $x^p$ is dominant
-- if $p \lt q$, the shape of $x^{1 \over q}$ is dominant
+- if $p > q$, the shape of $x^p$ is dominant
+- if $p < q$, the shape of $x^{1 \over q}$ is dominant
 - points $(0, 0)$ and $(1, 1)$ will always lie on graph
 - Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 
@@ -104,6 +110,17 @@ Closed circle - point not included
 
 Addition of linear piecewise graphs - add $y$-values at key points
 
-Product functions:  
+Product functions:
+
 - product will equal 0 if one of the functions is equal to 0
-- turning point on one function does not equate to turning point on product
\ No newline at end of file
+- turning point on one function does not equate to turning point on product
+
+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+
+## Composite functions
+
+$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
+
+