[spec] formatting for calculus rules
[notes.git] / methods / transformations-ref.tex
index 22532c4e956edfc46899efffbe165cacd69aacba..1f0390aac1c92519c5f3a275479e249b855a96d7 100644 (file)
@@ -1,7 +1,4 @@
-\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}
@@ -31,8 +28,6 @@
   \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
@@ -83,8 +75,7 @@
   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:
 
@@ -94,12 +85,10 @@ 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:
@@ -128,9 +117,7 @@ For graph of \(y={1 \over x}\), horizontal \& vertical dilations are
 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 \(y=Af[n(x+c)] + b\)
@@ -142,14 +129,12 @@ reflection across \(x\)-axis)\\
 \(c\) - 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 \(-f(x) = f(-x)\)
@@ -158,39 +143,19 @@ 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
 
-\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
@@ -203,9 +168,7 @@ Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd
 \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.
 
@@ -216,9 +179,7 @@ Domain is:
 
 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 \(p, q \in \mathbb{Z}^+\)}
 
 \[x^{p \over q} = \sqrt[q]{x^p}\]
 
@@ -235,18 +196,14 @@ If \(n\) is odd, it is an odd function.
   \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\)
 \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
@@ -273,14 +230,12 @@ Product functions:
   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
 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)