1\definecolor{shade1}{HTML}{ffffff} 2\definecolor{shade2}{HTML}{e6f2ff} 3\definecolor{shade3}{HTML}{cce2ff} 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 \(\parallel y\)-axis/from \(x\)-axis 21\item 22 translation of \(h\) units \(\parallel x\)-axis/from \(y\)-axis 23\item 24 for \((ax)^n\), dilation factor is \(1\over a \> \parallel x\)-axis/from \(y\)-axis 25\item 26 when \(0 < |a| < 1\), graph becomes closer to axis 27\end{itemize} 28 29\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)} 30 31Applies to exponential, log, trig, \(e^x\), polynomials.\\ 32Functions must be written in form \(y=Af[n(x+c)]+b\) 33 34\begin{itemize} 35\tightlist 36\item 37 dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection 38 across \(y\)-axis) 39\item 40 dilation by factor \(1\over n\) from \(y\)-axis (if \(n<0\), 41 reflection across \(x\)-axis) 42\item 43 translation of \(c\) units from \(y\)-axis (\(x\)-shift) 44\item 45 translation of \(b\) units from \(x\)-axis (\(y\)-shift) 46\end{itemize} 47 48\subsection*{Dilations} 49 50Two pairs of equivalent processes for \(y=f(x)\): 51 52\begin{enumerate} 53\def\labelenumi{\arabic{enumi}.} 54\item 55\begin{itemize} 56\tightlist 57\item 58 Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\) 59\item 60 Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\) 61\end{itemize} 62\item 63\begin{itemize} 64\tightlist 65\item 66 Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\) 67\item 68 Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\) 69\end{itemize} 70\end{enumerate} 71 72For graph of \(y={1\over x}\), horizontal \& vertical dilations are 73equivalent (symmetrical). If \(y={a \over x}\), graph is contracted 74rather than dilated. 75 76\subsection*{Matrix transformations} 77 78Find new point \((x^\prime, y^\prime)\). Substitute these into original 79equation to find image with original variables \((x, y)\). 80 81\subsection*{Reflections} 82 83\begin{itemize} 84\tightlist 85\item 86 Reflection \textbf{in} axis = reflection \textbf{over} axis = 87 reflection \textbf{across} axis 88\item 89 Translations do not change 90\end{itemize} 91 92\subsection*{Translations} 93 94For \(y = f(x)\), these processes are equivalent: 95 96\begin{itemize} 97\tightlist 98\item 99 applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the 100 graph of \(y = f(x)\) 101\item 102 replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain 103 \(y-k = f(x-h)\) 104\end{itemize} 105 106\subsection*{Power functions} 107 108Mostly only on CAS. 109 110We can write 111\(x^{-1\over n} = {1\over{x^{1\over n}}} = {1\over ^n \sqrt{x}}\)n.\\ 112Domain is: 113\(\begin{cases}\mathbb{R}\setminus \{0\}\hspace{0.5em}\text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\) 114 115If \(n\) is odd, it is an odd function. 116 117\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)} 118 119\[x^{p \over q} = \sqrt[q]{x^p}\] 120 121\begin{itemize} 122\tightlist 123\item 124 if \(p > q\), the shape of \(x^p\) is dominant 125\item 126 if \(p < q\), the shape of \(x^{1\over q}\) is dominant 127\item 128 points \((0, 0)\) and \((1, 1)\) will always lie on graph 129\item 130 Domain is: 131 \(\begin{cases}\mathbb{R}\hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\}\hspace{1em}\text{if }q\text{ is even}\end{cases}\) 132\end{itemize} 133