-# Transformation
+---
+geometry: a4paper, margin=2cm
+columns: 2
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+---
-**Order of operations:** DRT - Dilations, Reflections, Translations
+\setstretch{1.6}
+\pagenumbering{gobble}
+
+# Transformations
+
+**Order of operations:** DRT
+
+\begin{center}dilations --- reflections --- translations\end{center}
## Transforming $x^n$ to $a(x-h)^n+K$
-- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
+- dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
- if $a<0$, graph is reflected over $x$-axis
-- $k$ - 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
+- translation of $k$ units parallel to $y$-axis or from $x$-axis
+- 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
+## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
-For $y = f(x)$, these processes are equivalent:
+Applies to exponential, log, trig, $e^x$, polynomials.
+Functions must be written in form $y=Af[n(x+c)]+b$
-- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
-- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+- dilation by factor $|A|$ from $x$-axis (if $A<0$, reflection across $y$-axis)
+- dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
+- translation of $c$ units from $y$-axis ($x$-shift)
+- translation of $b$ units from $x$-axis ($y$-shift)
## Dilations
-For the graph of $y = f(x)$, there are two pairs of equivalent processes:
+Two pairs of equivalent processes for $y=f(x)$:
1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
- Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
-## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+
+## Reflections
+
+- Reflection **in** axis = reflection **over** axis = reflection **across** axis
+- Translations do not change
-Applies to exponential, log, trig, power, polynomial functions.
-Functions must be written in form $y=Af[n(x+c)] + b$
+## Translations
+
+For $y = f(x)$, these processes are equivalent:
-$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)
-$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
-$c$ - translation from $y$-axis ($x$-shift)
-$b$ - translation from $x$-axis ($y$-shift)
+- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
+- replacing $x$ with $x-h$ and $y$ with $y-k$ to obtain $y-k = f(x-h)$
## Power functions
Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$
Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
-### $x^n$ where $n \in \mathbb{Z}^+$
-
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
-
-### $x^n$ where $n \in \mathbb{Z}^-$
-
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
-
-### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
-
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
+\begin{center}
+\begin{tabular}{m{1.2cm}|C|C}
+ & $n$ is even & $n$ is odd \\
+ \hline
+ \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
+ \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
+ \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
+\end{tabular}
+\end{center}
### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
If $n$ is odd, it is an odd function.
+\columnbreak
+
### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
$$x^{p \over q} = \sqrt[q]{x^p}$$
-- 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: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
+## Piecewise functions
-## Combinations of functions (piecewise/hybrid)
+$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
-$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
+**Open circle:** point included
+**Closed circle:** point not included
-Open circle - point included
-Closed circle - point not included
+## Operations on functions
-### Sum, difference, product of functions
-| | | |
-|---|-----|-----|
-|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$|
-|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
-|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
+For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$
-Addition of linear piecewise graphs - add $y$-values at key points
+Addition of linear piecewise graphs: add $y$-values at key points
-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
+Product functions:
-## Matrix transformations
-
-Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+- product will equal 0 if $f=0$ or $g=0$
+- $f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0$
## Composite functions