methods / transformations-ref.texon commit [chem] start organic reactions summary document (5447e6c)
   1\documentclass[standalone]{article}
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  46
  47\author{Andrew Lorimer}
  48\date{}
  49
  50\begin{document}
  51
  52\section{Transformations}
  53
  54\textbf{Order of operations:} DRT - Dilations, Reflections, Translations
  55
  56\subsection{Transforming x\^{}n to a(x-h)\^{}n+K}
  57
  58\begin{itemize}
  59\tightlist
  60\item
  61  \(|a|\) is the dilation factor of \(|a|\) units parallel to \(y\)-axis
  62  or from \(x\)-axis
  63\item
  64  if \(a<0\), graph is reflected over \(x\)-axis
  65\item
  66  \(k\) - translation of \(k\) units parallel to \(y\)-axis or from
  67  \(x\)-axis
  68\item
  69  \(h\) - translation of \(h\) units parallel to \(x\)-axis or from
  70  \(y\)-axis
  71\item
  72  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
  73  \(x\)-axis or from \(y\)-axis
  74\item
  75  when \(0 < |a| < 1\), graph becomes closer to axis
  76\end{itemize}
  77
  78\subsection{Translations}
  79
  80For \(y = f(x)\), these processes are equivalent:
  81
  82\begin{itemize}
  83\tightlist
  84\item
  85  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
  86  graph of \(y = f(x)\)
  87\item
  88  replacing \(x\) with \(x - h\) and \(y\) with \(y - k\) to obtain \(y - k = f (x - h)\)
  89\end{itemize}
  90
  91\subsection{Dilations}
  92
  93For the graph of \(y = f(x)\), there are two pairs of equivalent
  94processes:
  95
  96\begin{enumerate}
  97\def\labelenumi{\arabic{enumi}.}
  98\item
  99  \begin{itemize}
 100  \tightlist
 101  \item
 102    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
 103  \item
 104    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
 105  \end{itemize}
 106\item
 107  \begin{itemize}
 108  \tightlist
 109  \item
 110    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
 111  \item
 112    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
 113  \end{itemize}
 114\end{enumerate}
 115
 116For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
 117equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
 118rather than dilated.
 119
 120\subsection{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
 121
 122Applies to exponential, log, trig, power, polynomial functions.\\
 123Functions must be written in form \(y=Af[n(x+c)] + b\)
 124
 125\(A\) - dilation by factor \(A\) from \(x\)-axis (if \(A<0\), reflection
 126across \(y\)-axis)\\
 127\(n\) - dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
 128reflection across \(x\)-axis)\\
 129\(c\) - translation from \(y\)-axis (\(x\)-shift)\\
 130\(b\) - translation from \(x\)-axis (\(y\)-shift)
 131
 132\subsection{Power functions}
 133
 134\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
 135(including \(x=0\))
 136
 137\subsubsection{Odd and even functions}
 138
 139Even when \(f(x) = -f(x)\)\\
 140Odd when \(-f(x) = f(-x)\)
 141
 142Function is even if it can be reflected across \(y\)-axis
 143\(\implies f(x)=f(-x)\)\\
 144Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
 145
 146\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
 147\begin{center}
 148\begin{tabular}{m{1.2cm}|C|C}
 149  & $n$ is even & $n$ is odd \\
 150  \hline
 151  \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
 152  \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
 153  \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}}\\
 154\end{tabular}
 155\end{center}
 156\subsubsection{\(x^n\) where \(n \in \mathbb{Z}^+\)}
 157
 158\subsubsection{\(x^{1 \over n}\) where \(n \in \mathbb{Z}^+\)}
 159
 160\begin{longtable}[]{@{}ll@{}}
 161\toprule
 162\(n\) is even: & \(n\) is odd:\tabularnewline
 163\midrule
 164\endhead
 165\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/square-root-graph.png}
 166&
 167\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cube-root-graph.png}\tabularnewline
 168\bottomrule
 169\end{longtable}
 170
 171\subsubsection{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}
 172
 173Mostly only on CAS.
 174
 175We can write
 176\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
 177Domain is:
 178\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
 179
 180If \(n\) is odd, it is an odd function.
 181
 182\subsubsection{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
 183
 184\[x^{p \over q} = \sqrt[q]{x^p}\]
 185
 186\begin{itemize}
 187\tightlist
 188\item
 189  if \(p > q\), the shape of \(x^p\) is dominant
 190\item
 191  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
 192\item
 193  points \((0, 0)\) and \((1, 1)\) will always lie on graph
 194\item
 195  Domain is:
 196  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 197\end{itemize}
 198
 199\subsection{Combinations of functions (piecewise/hybrid)}
 200
 201\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 202
 203Open circle - point included\\
 204Closed circle - point not included
 205
 206\subsubsection{Sum, difference, product of functions}
 207
 208\begin{longtable}[]{@{}lll@{}}
 209\toprule
 210\endhead
 211sum & \(f+g\) & domain
 212\(= \text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 213difference & \(f-g\) or \(g-f\) & domain
 214\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 215product & \(f \times g\) & domain
 216\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 217\bottomrule
 218\end{longtable}
 219
 220Addition of linear piecewise graphs - add \(y\)-values at key points
 221
 222Product functions:
 223
 224\begin{itemize}
 225\tightlist
 226\item
 227  product will equal 0 if one of the functions is equal to 0
 228\item
 229  turning point on one function does not equate to turning point on
 230  product
 231\end{itemize}
 232
 233\subsection{Matrix transformations}
 234
 235Find new point \((x^\prime, y^\prime)\). Substitute these into original
 236equation to find image with original variables \((x, y)\).
 237
 238\subsection{Composite functions}
 239
 240\((f \circ g)(x)\) is defined iff
 241\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 242
 243\end{document}