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\author{Andrew Lorimer}
\date{}

\begin{document}

\section{Transformations}

\textbf{Order of operations:} DRT - Dilations, Reflections, Translations

\subsection{Transforming x\^{}n to a(x-h)\^{}n+K}

\begin{itemize}
\tightlist
\item
  \(|a|\) is the dilation factor of \(|a|\) units parallel to \(y\)-axis
  or from \(x\)-axis
\item
  if \(a<0\), graph is reflected over \(x\)-axis
\item
  \(k\) - translation of \(k\) units parallel to \(y\)-axis or from
  \(x\)-axis
\item
  \(h\) - translation of \(h\) units parallel to \(x\)-axis or from
  \(y\)-axis
\item
  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
  \(x\)-axis or from \(y\)-axis
\item
  when \(0 < |a| < 1\), graph becomes closer to axis
\end{itemize}

\subsection{Translations}

For \(y = f(x)\), these processes are equivalent:

\begin{itemize}
\tightlist
\item
  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
  graph of \(y = f(x)\)
\item
  replacing \(x\) with \(x - h\) and \(y\) with \(y - k\) to obtain \(y - k = f (x - h)\)
\end{itemize}

\subsection{Dilations}

For the graph of \(y = f(x)\), there are two pairs of equivalent
processes:

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \begin{itemize}
  \tightlist
  \item
    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
  \item
    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
  \end{itemize}
\item
  \begin{itemize}
  \tightlist
  \item
    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
  \item
    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
  \end{itemize}
\end{enumerate}

For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
rather than dilated.

\subsection{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}

Applies to exponential, log, trig, power, polynomial functions.\\
Functions must be written in form \(y=Af[n(x+c)] + b\)

\(A\) - 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)

\subsection{Power functions}

\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
(including \(x=0\))

\subsubsection{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

\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}
\subsubsection{\(x^n\) where \(n \in \mathbb{Z}^+\)}

\subsubsection{\(x^{1 \over n}\) where \(n \in \mathbb{Z}^+\)}

\begin{longtable}[]{@{}ll@{}}
\toprule
\(n\) is even: & \(n\) is odd:\tabularnewline
\midrule
\endhead
\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/square-root-graph.png}
&
\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cube-root-graph.png}\tabularnewline
\bottomrule
\end{longtable}

\subsubsection{\(x^{-1 \over n}\) 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.

\subsubsection{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}

\[x^{p \over q} = \sqrt[q]{x^p}\]

\begin{itemize}
\tightlist
\item
  if \(p > q\), the shape of \(x^p\) is dominant
\item
  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
\item
  points \((0, 0)\) and \((1, 1)\) will always lie on graph
\item
  Domain is:
  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
\end{itemize}

\subsection{Combinations of functions (piecewise/hybrid)}

\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]

Open circle - point included\\
Closed circle - point not included

\subsubsection{Sum, difference, product of functions}

\begin{longtable}[]{@{}lll@{}}
\toprule
\endhead
sum & \(f+g\) & domain
\(= \text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
difference & \(f-g\) or \(g-f\) & domain
\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
product & \(f \times g\) & domain
\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
\bottomrule
\end{longtable}

Addition of linear piecewise graphs - add \(y\)-values at key points

Product functions:

\begin{itemize}
\tightlist
\item
  product will equal 0 if one of the functions is equal to 0
\item
  turning point on one function does not equate to turning point on
  product
\end{itemize}

\subsection{Matrix transformations}

Find new point \((x^\prime, y^\prime)\). Substitute these into original
equation to find image with original variables \((x, y)\).

\subsection{Composite functions}

\((f \circ g)(x)\) is defined iff
\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

\end{document}
