methods / transformations-ref.texon commit [english] finish plan for oral presentation (3b02698)
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  50
  51
  52\author{Andrew Lorimer}
  53\date{}
  54
  55\begin{document}
  56
  57\hypertarget{transformation}{%
  58\section{Transformation}\label{transformation}}
  59
  60\textbf{Order of operations:} DRT - Dilations, Reflections, Translations
  61
  62\hypertarget{transforming-xn-to-ax-hnk}{%
  63\subsection{\texorpdfstring{Transforming \(x^n\) to
  64\(a(x-h)^n+K\)}{Transforming x\^{}n to a(x-h)\^{}n+K}}\label{transforming-xn-to-ax-hnk}}
  65
  66\begin{itemize}
  67\tightlist
  68\item
  69  \(|a|\) is the dilation factor of \(|a|\) units parallel to \(y\)-axis
  70  or from \(x\)-axis
  71\item
  72  if \(a<0\), graph is reflected over \(x\)-axis
  73\item
  74  \(k\) - translation of \(k\) units parallel to \(y\)-axis or from
  75  \(x\)-axis
  76\item
  77  \(h\) - translation of \(h\) units parallel to \(x\)-axis or from
  78  \(y\)-axis
  79\item
  80  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
  81  \(x\)-axis or from \(y\)-axis
  82\item
  83  when \(0 < |a| < 1\), graph becomes closer to axis
  84\end{itemize}
  85
  86\hypertarget{translations}{%
  87\subsection{Translations}\label{translations}}
  88
  89For \(y = f(x)\), these processes are equivalent:
  90
  91\begin{itemize}
  92\tightlist
  93\item
  94  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
  95  graph of \(y = f(x)\)
  96\item
  97  replacing \(x\) with \(x − h\) and \(y\) with \(y − k\) to obtain
  98  \(y − k = f (x − h)\)
  99\end{itemize}
 100
 101\hypertarget{dilations}{%
 102\subsection{Dilations}\label{dilations}}
 103
 104For the graph of \(y = f(x)\), there are two pairs of equivalent
 105processes:
 106
 107\begin{enumerate}
 108\def\labelenumi{\arabic{enumi}.}
 109\item
 110  \begin{itemize}
 111  \tightlist
 112  \item
 113    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
 114  \item
 115    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
 116  \end{itemize}
 117\item
 118  \begin{itemize}
 119  \tightlist
 120  \item
 121    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
 122  \item
 123    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
 124  \end{itemize}
 125\end{enumerate}
 126
 127For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
 128equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
 129rather than dilated.
 130
 131\hypertarget{transforming-fx-to-yafnxcb}{%
 132\subsection{\texorpdfstring{Transforming \(f(x)\) to
 133\(y=Af[n(x+c)]+b\)}{Transforming f(x) to y=Af{[}n(x+c){]}+b}}\label{transforming-fx-to-yafnxcb}}
 134
 135Applies to exponential, log, trig, power, polynomial functions.\\
 136Functions must be written in form \(y=Af[n(x+c)] + b\)
 137
 138\(A\) - dilation by factor \(A\) from \(x\)-axis (if \(A<0\), reflection
 139across \(y\)-axis)\\
 140\(n\) - dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
 141reflection across \(x\)-axis)\\
 142\(c\) - translation from \(y\)-axis (\(x\)-shift)\\
 143\(b\) - translation from \(x\)-axis (\(y\)-shift)
 144
 145\hypertarget{power-functions}{%
 146\subsection{Power functions}\label{power-functions}}
 147
 148\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
 149(including \(x=0\))
 150
 151\hypertarget{odd-and-even-functions}{%
 152\subsubsection{Odd and even functions}\label{odd-and-even-functions}}
 153
 154Even when \(f(x) = -f(x)\)\\
 155Odd when \(-f(x) = f(-x)\)
 156
 157Function is even if it can be reflected across \(y\)-axis
 158\(\implies f(x)=f(-x)\)\\
 159Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
 160
 161\hypertarget{xn-where-n-in-mathbbz}{%
 162\subsubsection{\texorpdfstring{\(x^n\) where
 163\(n \in \mathbb{Z}^+\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xn-where-n-in-mathbbz}}
 164
 165\begin{longtable}[]{@{}ll@{}}
 166\toprule
 167\(n\) is even: & \(n\) is odd:\tabularnewline
 168\midrule
 169\endhead
 170\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/parabola.png}
 171&
 172\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cubic.png}\tabularnewline
 173\bottomrule
 174\end{longtable}
 175
 176\hypertarget{xn-where-n-in-mathbbz-}{%
 177\subsubsection{\texorpdfstring{\(x^n\) where
 178\(n \in \mathbb{Z}^-\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}-}}\label{xn-where-n-in-mathbbz-}}
 179
 180\begin{longtable}[]{@{}ll@{}}
 181\toprule
 182\(n\) is even: & \(n\) is odd:\tabularnewline
 183\midrule
 184\endhead
 185\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/truncus.png}
 186&
 187\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/hyperbola.png}\tabularnewline
 188\bottomrule
 189\end{longtable}
 190
 191\hypertarget{x1-over-n-where-n-in-mathbbz}{%
 192\subsubsection{\texorpdfstring{\(x^{1 \over n}\) where
 193\(n \in \mathbb{Z}^+\)}{x\^{}\{1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x1-over-n-where-n-in-mathbbz}}
 194
 195\begin{longtable}[]{@{}ll@{}}
 196\toprule
 197\(n\) is even: & \(n\) is odd:\tabularnewline
 198\midrule
 199\endhead
 200\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/square-root-graph.png}
 201&
 202\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cube-root-graph.png}\tabularnewline
 203\bottomrule
 204\end{longtable}
 205
 206\hypertarget{x-1-over-n-where-n-in-mathbbz}{%
 207\subsubsection{\texorpdfstring{\(x^{-1 \over n}\) where
 208\(n \in \mathbb{Z}^+\)}{x\^{}\{-1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x-1-over-n-where-n-in-mathbbz}}
 209
 210Mostly only on CAS.
 211
 212We can write
 213\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
 214Domain is:
 215\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
 216
 217If \(n\) is odd, it is an odd function.
 218
 219\hypertarget{xp-over-q-where-p-q-in-mathbbz}{%
 220\subsubsection{\texorpdfstring{\(x^{p \over q}\) where
 221\(p, q \in \mathbb{Z}^+\)}{x\^{}\{p \textbackslash{}over q\} where p, q \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xp-over-q-where-p-q-in-mathbbz}}
 222
 223\[x^{p \over q} = \sqrt[q]{x^p}\]
 224
 225\begin{itemize}
 226\tightlist
 227\item
 228  if \(p > q\), the shape of \(x^p\) is dominant
 229\item
 230  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
 231\item
 232  points \((0, 0)\) and \((1, 1)\) will always lie on graph
 233\item
 234  Domain is:
 235  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 236\end{itemize}
 237
 238\hypertarget{combinations-of-functions-piecewisehybrid}{%
 239\subsection{Combinations of functions
 240(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}}
 241
 242\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 243
 244Open circle - point included\\
 245Closed circle - point not included
 246
 247\hypertarget{sum-difference-product-of-functions}{%
 248\subsubsection{Sum, difference, product of
 249functions}\label{sum-difference-product-of-functions}}
 250
 251\begin{longtable}[]{@{}lll@{}}
 252\toprule
 253\endhead
 254sum & \(f+g\) & domain
 255\(= \text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 256difference & \(f-g\) or \(g-f\) & domain
 257\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 258product & \(f \times g\) & domain
 259\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
 260\bottomrule
 261\end{longtable}
 262
 263Addition of linear piecewise graphs - add \(y\)-values at key points
 264
 265Product functions:
 266
 267\begin{itemize}
 268\tightlist
 269\item
 270  product will equal 0 if one of the functions is equal to 0
 271\item
 272  turning point on one function does not equate to turning point on
 273  product
 274\end{itemize}
 275
 276\hypertarget{matrix-transformations}{%
 277\subsection{Matrix transformations}\label{matrix-transformations}}
 278
 279Find new point \((x^\prime, y^\prime)\). Substitute these into original
 280equation to find image with original variables \((x, y)\).
 281
 282\hypertarget{composite-functions}{%
 283\subsection{Composite functions}\label{composite-functions}}
 284
 285\((f \circ g)(x)\) is defined iff
 286\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 287
 288\end{document}