-\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
-\PassOptionsToPackage{hyphens}{url}
-%
-\documentclass[]{article}
+\documentclass[standalone]{article}
\usepackage{lmodern}
\usepackage{amssymb,amsmath}
\usepackage{ifxetex,ifluatex}
\KOMAoptions{parskip=half}}
\makeatother
\usepackage{xcolor}
-\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available
-\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}}
\urlstyle{same} % don't use monospace font for urls
\usepackage{fullpage}
\usepackage{longtable,booktabs}
\begin{document}
-\hypertarget{transformation}{%
-\section{Transformation}\label{transformation}}
+\section{Transformations}
\textbf{Order of operations:} DRT - Dilations, Reflections, Translations
-\hypertarget{transforming-xn-to-ax-hnk}{%
-\subsection{\texorpdfstring{Transforming \(x^n\) to
-\(a(x-h)^n+K\)}{Transforming x\^{}n to a(x-h)\^{}n+K}}\label{transforming-xn-to-ax-hnk}}
+\subsection{Transforming x\^{}n to a(x-h)\^{}n+K}
\begin{itemize}
\tightlist
when \(0 < |a| < 1\), graph becomes closer to axis
\end{itemize}
-\hypertarget{translations}{%
-\subsection{Translations}\label{translations}}
+\subsection{Translations}
For \(y = f(x)\), these processes are equivalent:
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
- \(y − k = f (x − h)\)
+ replacing \(x\) with \(x - h\) and \(y\) with \(y - k\) to obtain \(y - k = f (x - h)\)
\end{itemize}
-\hypertarget{dilations}{%
-\subsection{Dilations}\label{dilations}}
+\subsection{Dilations}
For the graph of \(y = f(x)\), there are two pairs of equivalent
processes:
equivalent (symmetrical). If \(y={a \over x}\), graph is contracted
rather than dilated.
-\hypertarget{transforming-fx-to-yafnxcb}{%
-\subsection{\texorpdfstring{Transforming \(f(x)\) to
-\(y=Af[n(x+c)]+b\)}{Transforming f(x) to y=Af{[}n(x+c){]}+b}}\label{transforming-fx-to-yafnxcb}}
+\subsection{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)}
Applies to exponential, log, trig, power, polynomial functions.\\
Functions must be written in form \(y=Af[n(x+c)] + b\)
\(c\) - translation from \(y\)-axis (\(x\)-shift)\\
\(b\) - translation from \(x\)-axis (\(y\)-shift)
-\hypertarget{power-functions}{%
-\subsection{Power functions}\label{power-functions}}
+\subsection{Power functions}
\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\)
(including \(x=0\))
-\hypertarget{odd-and-even-functions}{%
-\subsubsection{Odd and even functions}\label{odd-and-even-functions}}
+\subsubsection{Odd and even functions}
Even when \(f(x) = -f(x)\)\\
Odd when \(-f(x) = f(-x)\)
\(\implies f(x)=f(-x)\)\\
Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
-\hypertarget{xn-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^n\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xn-where-n-in-mathbbz}}
+\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
+\begin{center}
+\begin{tabular}{m{1.2cm}|C|C}
+ & $n$ is even & $n$ is odd \\
+ \hline
+ \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
+ \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
+ \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
+\end{tabular}
+\end{center}
+\subsubsection{\(x^n\) where \(n \in \mathbb{Z}^+\)}
-\begin{longtable}[]{@{}ll@{}}
-\toprule
-\(n\) is even: & \(n\) is odd:\tabularnewline
-\midrule
-\endhead
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/parabola.png}
-&
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cubic.png}\tabularnewline
-\bottomrule
-\end{longtable}
-
-\hypertarget{xn-where-n-in-mathbbz-}{%
-\subsubsection{\texorpdfstring{\(x^n\) where
-\(n \in \mathbb{Z}^-\)}{x\^{}n where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}-}}\label{xn-where-n-in-mathbbz-}}
-
-\begin{longtable}[]{@{}ll@{}}
-\toprule
-\(n\) is even: & \(n\) is odd:\tabularnewline
-\midrule
-\endhead
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/truncus.png}
-&
-\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/hyperbola.png}\tabularnewline
-\bottomrule
-\end{longtable}
-
-\hypertarget{x1-over-n-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{1 \over n}\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}\{1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x1-over-n-where-n-in-mathbbz}}
+\subsubsection{\(x^{1 \over n}\) where \(n \in \mathbb{Z}^+\)}
\begin{longtable}[]{@{}ll@{}}
\toprule
\bottomrule
\end{longtable}
-\hypertarget{x-1-over-n-where-n-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{-1 \over n}\) where
-\(n \in \mathbb{Z}^+\)}{x\^{}\{-1 \textbackslash{}over n\} where n \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{x-1-over-n-where-n-in-mathbbz}}
+\subsubsection{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)}
Mostly only on CAS.
If \(n\) is odd, it is an odd function.
-\hypertarget{xp-over-q-where-p-q-in-mathbbz}{%
-\subsubsection{\texorpdfstring{\(x^{p \over q}\) where
-\(p, q \in \mathbb{Z}^+\)}{x\^{}\{p \textbackslash{}over q\} where p, q \textbackslash{}in \textbackslash{}mathbb\{Z\}\^{}+}}\label{xp-over-q-where-p-q-in-mathbbz}}
+\subsubsection{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)}
\[x^{p \over q} = \sqrt[q]{x^p}\]
\(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
\end{itemize}
-\hypertarget{combinations-of-functions-piecewisehybrid}{%
-\subsection{Combinations of functions
-(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}}
+\subsection{Combinations of functions (piecewise/hybrid)}
\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
Open circle - point included\\
Closed circle - point not included
-\hypertarget{sum-difference-product-of-functions}{%
-\subsubsection{Sum, difference, product of
-functions}\label{sum-difference-product-of-functions}}
+\subsubsection{Sum, difference, product of functions}
\begin{longtable}[]{@{}lll@{}}
\toprule
product
\end{itemize}
-\hypertarget{matrix-transformations}{%
-\subsection{Matrix transformations}\label{matrix-transformations}}
+\subsection{Matrix transformations}
Find new point \((x^\prime, y^\prime)\). Substitute these into original
equation to find image with original variables \((x, y)\).
-\hypertarget{composite-functions}{%
-\subsection{Composite functions}\label{composite-functions}}
+\subsection{Composite functions}
\((f \circ g)(x)\) is defined iff
\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)