+---
+geometry: margin=2cm
+columns: 2
+author: Andrew Lorimer
+header-includes:
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+---
+
# Transformation
**Order of operations:** DRT - Dilations, Reflections, Translations
-## $f(x) = x^n$ to $f(x)=a(x-h)^n+K$##
+## 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
- 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
-
-## Translations
-
-For $y = f(x)$, these processes are equivalent:
-
-- 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 $y − k = f (x − h)$
+- 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
## Dilations
For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
-## Transformations from $f(x)$ to $y=Af[n(x+c)]+b$#
+## Reflections
+
+- Reflection **in** axis = reflection **over** axis = reflection **across** axis
+- Translations do not change
+
+## Translations
+
+For $y = f(x)$, these processes are equivalent:
+
+- 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 $y − k = f (x − h)$
+
+## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
Applies to exponential, log, trig, power, polynomial functions.
Functions must be written in form $y=Af[n(x+c)] + b$
## Power functions
-**Strictly increasing** on an interval where $x_2 > x_1 \implies f(x_2) > f(x_2)$ (including $x=0$)
+**Strictly increasing:** $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$)
+
+### Odd and even functions
+Even when $f(x) = -f(x)$
+Odd when $-f(x) = f(-x)$
+
+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 $n \in \mathbb{Z}^+$
+
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/parabola.png}} & {\includegraphics[height=1cm]{graphics/cubic.png}}
+\end{tabularx}
+
+### $x^n$ where $n \in \mathbb{Z}^-$
+
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/truncus.png}} & {\includegraphics[height=1cm]{graphics/hyperbola.png}}
+\end{tabularx}
+
+### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
+
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/square-root-graph.png}} & {\includegraphics[height=1cm]{graphics/cube-root-graph.png}}
+\end{tabularx}
+
+### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
+
+Mostly only on CAS.
+
+We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n.
+Domain is: $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
+
+If $n$ is odd, it is an odd function.
+
+### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
+
+$$x^{p \over q} = \sqrt[q]{x^p}$$
+
+- 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: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
+
+
+## Combinations of functions (piecewise/hybrid)
+
+$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
+
+Open circle - point included
+Closed circle - point not included
+
+### Sum, difference, product of functions
+\begin{tabularx}{\columnwidth}{X|X}
+ 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 $=\text{dom}(f) \cap \text{dom}(g)$
+\end{tabularx}
+
+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
+
+## 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 $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
-#### $n$ is odd and $n>1$:
-$f(-x)=-f(x)$
-#### $n$ is even and $n>1$:
-$f(-x)=f(x)$
\ No newline at end of file