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\author{Andrew Lorimer}
\date{}

\begin{document}

\hypertarget{transformation}{%
\section{Transformation}\label{transformation}}

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

\hypertarget{transforming-xn-to-ax-hnk}{%
\subsection{\texorpdfstring{Transforming \(x^n\) to
\(a(x-h)^n+K\)}{Transforming x\^{}n to a(x-h)\^{}n+K}}\label{transforming-xn-to-ax-hnk}}

\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}

\hypertarget{translations}{%
\subsection{Translations}\label{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}

\hypertarget{dilations}{%
\subsection{Dilations}\label{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.

\hypertarget{transforming-fx-to-yafnxcb}{%
\subsection{\texorpdfstring{Transforming \(f(x)\) to
\(y=Af[n(x+c)]+b\)}{Transforming f(x) to y=Af{[}n(x+c){]}+b}}\label{transforming-fx-to-yafnxcb}}

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)

\hypertarget{power-functions}{%
\subsection{Power functions}\label{power-functions}}

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

\hypertarget{odd-and-even-functions}{%
\subsubsection{Odd and even functions}\label{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

\hypertarget{xn-where-n-in-mathbbz}{%
\subsubsection{\texorpdfstring{\(x^n\) where
\(n \in \mathbb{Z}^+\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xn-where-n-in-mathbbz}}

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

\hypertarget{xn-where-n-in-mathbbz-}{%
\subsubsection{\texorpdfstring{\(x^n\) where
\(n \in \mathbb{Z}^-\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}-}}\label{xn-where-n-in-mathbbz-}}

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

\hypertarget{x1-over-n-where-n-in-mathbbz}{%
\subsubsection{\texorpdfstring{\(x^{1 \over n}\) where
\(n \in \mathbb{Z}^+\)}{x\^{}\{1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x1-over-n-where-n-in-mathbbz}}

\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}

\hypertarget{x-1-over-n-where-n-in-mathbbz}{%
\subsubsection{\texorpdfstring{\(x^{-1 \over n}\) where
\(n \in \mathbb{Z}^+\)}{x\^{}\{-1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x-1-over-n-where-n-in-mathbbz}}

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.

\hypertarget{xp-over-q-where-p-q-in-mathbbz}{%
\subsubsection{\texorpdfstring{\(x^{p \over q}\) where
\(p, q \in \mathbb{Z}^+\)}{x\^{}\{p \textbackslash{}over q\} where p, q \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xp-over-q-where-p-q-in-mathbbz}}

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

\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}

\hypertarget{combinations-of-functions-piecewisehybrid}{%
\subsection{Combinations of functions
(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}}

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

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

\hypertarget{sum-difference-product-of-functions}{%
\subsubsection{Sum, difference, product of
functions}\label{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}

\hypertarget{matrix-transformations}{%
\subsection{Matrix transformations}\label{matrix-transformations}}

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

\hypertarget{composite-functions}{%
\subsection{Composite functions}\label{composite-functions}}

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

\end{document}