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\section{Transformations}

\textbf{Order of operations:} DRT

\begin{center}dilations --- reflections --- translations\end{center}

\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)}

\begin{itemize}
\tightlist
\item
  dilation factor of \(|a|\) units parallel to \(y\)-axis or from
  \(x\)-axis
\item
  if \(a<0\), graph is reflected over \(x\)-axis
\item
  translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis
\item
  translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis
\item
  for \((ax)^n\), dilation factor is \(1 \over a\) parallel to
  \(x\)-axis or from \(y\)-axis
\item
  when \(0 < |a| < 1\), graph becomes closer to axis
\end{itemize}

\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}

Applies to exponential, log, trig, \(e^x\), polynomials.\\
Functions must be written in form \(y=Af[n(x+c)]+b\)

\begin{itemize}
\tightlist
\item
  dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection
  across \(y\)-axis)
\item
  dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
  reflection across \(x\)-axis)
\item
  translation of \(c\) units from \(y\)-axis (\(x\)-shift)
\item
  translation of \(b\) units from \(x\)-axis (\(y\)-shift)
\end{itemize}

\subsection*{Dilations}

Two pairs of equivalent processes for \(y=f(x)\):

\begin{enumerate}
\def\labelenumi{\arabic{enumi}.}
\item
  \begin{itemize}
  \tightlist
  \item
    Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\)
  \item
    Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\)
  \end{itemize}
\item
  \begin{itemize}
  \tightlist
  \item
    Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\)
  \item
    Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\)
  \end{itemize}
\end{enumerate}

For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
rather than dilated.

\subsection*{Matrix transformations}

Find new point \((x^\prime, y^\prime)\). Substitute these into original
equation to find image with original variables \((x, y)\).

\subsection*{Reflections}

\begin{itemize}
\tightlist
\item
  Reflection \textbf{in} axis = reflection \textbf{over} axis =
  reflection \textbf{across} axis
\item
  Translations do not change
\end{itemize}

\subsection*{Translations}

For \(y = f(x)\), these processes are equivalent:

\begin{itemize}
\tightlist
\item
  applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the
  graph of \(y = f(x)\)
\item
  replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain
  \(y-k = f(x-h)\)
\end{itemize}

\subsection*{Power functions}

\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
(including \(x=0\))

\subsubsection*{Odd and even functions}

Even when \(f(x) = -f(x)\)\\
Odd when \(-f(x) = f(-x)\)

Function is even if it can be reflected across \(y\)-axis
\(\implies f(x)=f(-x)\)\\
Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
                  

\subsubsection*{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}

Mostly only on CAS.

We can write
\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\
Domain is:
\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\)

If \(n\) is odd, it is an odd function.

\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}

\[x^{p \over q} = \sqrt[q]{x^p}\]

\begin{itemize}
\tightlist
\item
  if \(p > q\), the shape of \(x^p\) is dominant
\item
  if \(p < q\), the shape of \(x^{1 \over q}\) is dominant
\item
  points \((0, 0)\) and \((1, 1)\) will always lie on graph
\item
  Domain is:
  \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
\end{itemize}

\subsection*{Piecewise functions}

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

\textbf{Open circle:} point included\\
\textbf{Closed circle:} point not included

\subsection*{Operations on functions}

For \(f \pm g\) and \(f \times g\):
\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)

Addition of linear piecewise graphs: add \(y\)-values at key points

Product functions:

\begin{itemize}
\tightlist
\item
  product will equal 0 if \(f=0\) or \(g=0\)
\item
  \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
\end{itemize}

\subsection*{Composite functions}

\((f \circ g)(x)\) is defined iff
\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)