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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
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\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
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\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)