methods / transformations.texon commit [chem] start organic reactions summary document (5447e6c)
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   4\section{Transformations}
   5
   6\textbf{Order of operations:} DRT
   7
   8\begin{center}dilations --- reflections --- translations\end{center}
   9
  10\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)}
  11
  12\begin{itemize}
  13\tightlist
  14\item
  15  dilation factor of \(|a|\) units parallel to \(y\)-axis or from
  16  \(x\)-axis
  17\item
  18  if \(a<0\), graph is reflected over \(x\)-axis
  19\item
  20  translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis
  21\item
  22  translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis
  23\item
  24  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
  25  \(x\)-axis or from \(y\)-axis
  26\item
  27  when \(0 < |a| < 1\), graph becomes closer to axis
  28\end{itemize}
  29
  30\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
  31
  32Applies to exponential, log, trig, \(e^x\), polynomials.\\
  33Functions must be written in form \(y=Af[n(x+c)]+b\)
  34
  35\begin{itemize}
  36\tightlist
  37\item
  38  dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection
  39  across \(y\)-axis)
  40\item
  41  dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
  42  reflection across \(x\)-axis)
  43\item
  44  translation of \(c\) units from \(y\)-axis (\(x\)-shift)
  45\item
  46  translation of \(b\) units from \(x\)-axis (\(y\)-shift)
  47\end{itemize}
  48
  49\subsection*{Dilations}
  50
  51Two pairs of equivalent processes for \(y=f(x)\):
  52
  53\begin{enumerate}
  54\def\labelenumi{\arabic{enumi}.}
  55\item
  56  \begin{itemize}
  57  \tightlist
  58  \item
  59    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
  60  \item
  61    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
  62  \end{itemize}
  63\item
  64  \begin{itemize}
  65  \tightlist
  66  \item
  67    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
  68  \item
  69    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
  70  \end{itemize}
  71\end{enumerate}
  72
  73For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
  74equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
  75rather than dilated.
  76
  77\subsection*{Matrix transformations}
  78
  79Find new point \((x^\prime, y^\prime)\). Substitute these into original
  80equation to find image with original variables \((x, y)\).
  81
  82\subsection*{Reflections}
  83
  84\begin{itemize}
  85\tightlist
  86\item
  87  Reflection \textbf{in} axis = reflection \textbf{over} axis =
  88  reflection \textbf{across} axis
  89\item
  90  Translations do not change
  91\end{itemize}
  92
  93\subsection*{Translations}
  94
  95For \(y = f(x)\), these processes are equivalent:
  96
  97\begin{itemize}
  98\tightlist
  99\item
 100  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
 101  graph of \(y = f(x)\)
 102\item
 103  replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain
 104  \(y-k = f(x-h)\)
 105\end{itemize}
 106
 107\subsection*{Power functions}
 108
 109Mostly only on CAS.
 110
 111We can write
 112\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
 113Domain is:
 114\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)
 115
 116If \(n\) is odd, it is an odd function.
 117
 118\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
 119
 120\[x^{p \over q} = \sqrt[q]{x^p}\]
 121
 122\begin{itemize}
 123\tightlist
 124\item
 125  if \(p > q\), the shape of \(x^p\) is dominant
 126\item
 127  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
 128\item
 129  points \((0, 0)\) and \((1, 1)\) will always lie on graph
 130\item
 131  Domain is:
 132  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 133\end{itemize}
 134