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 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} 108 109\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\) 110(including \(x=0\)) 111 112 113\subsubsection*{\(x^{-1\over n}\) where \(n \in \mathbb{Z}^+\)} 114 115Mostly only on CAS. 116 117We can write 118\(x^{-1\over n} = {1\over{x^{1\over n}}} = {1\over ^n \sqrt{x}}\)n.\\ 119Domain is: 120\(\begin{cases}\mathbb{R}\setminus \{0\}\hspace{0.5em}\text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\) 121 122If \(n\) is odd, it is an odd function. 123 124\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)} 125 126\[x^{p \over q} = \sqrt[q]{x^p}\] 127 128\begin{itemize} 129\tightlist 130\item 131 if \(p > q\), the shape of \(x^p\) is dominant 132\item 133 if \(p < q\), the shape of \(x^{1\over q}\) is dominant 134\item 135 points \((0, 0)\) and \((1, 1)\) will always lie on graph 136\item 137 Domain is: 138 \(\begin{cases}\mathbb{R}\hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\}\hspace{1em}\text{if }q\text{ is even}\end{cases}\) 139\end{itemize} 140 141\subsection*{Piecewise functions} 142 143\[\text{e.g.}\quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le0 \\ 2, \hspace{3.4em}0 < x < 2 \\ x, \hspace{3.4em} x \ge2\end{cases}\] 144 145\textbf{Open circle:} point included\\ 146\textbf{Closed circle:} point not included 147 148\subsection*{Operations on functions} 149 150For \(f \pm g\) and \(f \times g\): 151\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\) 152 153Addition of linear piecewise graphs: add \(y\)-values at key points 154 155Product functions: 156 157\begin{itemize} 158\tightlist 159\item 160 product will equal 0 if \(f=0\) or \(g=0\) 161\item 162 \(f^\prime(x)=0\veebar g^\prime(x)=0\not\Rightarrow (f \times g)^\prime(x)=0\) 163\end{itemize} 164 165\subsection*{Composite functions} 166 167\((f \circ g)(x)\) is defined iff 168\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)