From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Sun, 3 Mar 2019 08:16:31 +0000 (+1100)
Subject: add transformations-ref source
X-Git-Tag: yr12~232
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/bd487b9356352199ce02b9c512ce5352379dff92

add transformations-ref source
---

diff --git a/methods/transformations-ref.tex b/methods/transformations-ref.tex
new file mode 100644
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+++ b/methods/transformations-ref.tex
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+\PassOptionsToPackage{unicode=true}{hyperref} % options for packages loaded elsewhere
+\PassOptionsToPackage{hyphens}{url}
+%
+\documentclass[]{article}
+\usepackage{lmodern}
+\usepackage{amssymb,amsmath}
+\usepackage{ifxetex,ifluatex}
+\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex
+  \usepackage[T1]{fontenc}
+  \usepackage[utf8]{inputenc}
+  \usepackage{textcomp} % provides euro and other symbols
+\else % if luatex or xelatex
+  \usepackage{unicode-math}
+  \defaultfontfeatures{Scale=MatchLowercase}
+  \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1}
+\fi
+% use upquote if available, for straight quotes in verbatim environments
+\IfFileExists{upquote.sty}{\usepackage{upquote}}{}
+\IfFileExists{microtype.sty}{% use microtype if available
+  \usepackage[]{microtype}
+  \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts
+}{}
+\makeatletter
+\@ifundefined{KOMAClassName}{% if non-KOMA class
+  \IfFileExists{parskip.sty}{%
+    \usepackage{parskip}
+  }{% else
+    \setlength{\parindent}{0pt}
+    \setlength{\parskip}{6pt plus 2pt minus 1pt}}
+}{% if KOMA class
+  \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}
+% Allow footnotes in longtable head/foot
+\IfFileExists{footnotehyper.sty}{\usepackage{footnotehyper}}{\usepackage{footnote}}
+\makesavenoteenv{longtable}
+\usepackage{graphicx,grffile}
+\makeatletter
+\makeatother
+
+% set default figure placement to htbp
+\makeatletter
+\def\fps@figure{htbp}
+\makeatother
+
+
+\author{Andrew Lorimer}
+\date{}
+
+\begin{document}
+
+\hypertarget{transformation}{%
+\section{Transformation}\label{transformation}}
+
+\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}}
+
+\begin{itemize}
+\tightlist
+\item
+  \(|a|\) is the dilation factor of \(|a|\) units parallel to \(y\)-axis
+  or from \(x\)-axis
+\item
+  if \(a<0\), graph is reflected over \(x\)-axis
+\item
+  \(k\) - translation of \(k\) units parallel to \(y\)-axis or from
+  \(x\)-axis
+\item
+  \(h\) - 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}
+
+\hypertarget{translations}{%
+\subsection{Translations}\label{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}
+
+\hypertarget{dilations}{%
+\subsection{Dilations}\label{dilations}}
+
+For the graph of \(y = f(x)\), there are two pairs of equivalent
+processes:
+
+\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.
+
+\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}}
+
+Applies to exponential, log, trig, power, polynomial functions.\\
+Functions must be written in form \(y=Af[n(x+c)] + b\)
+
+\(A\) - dilation by factor \(A\) from \(x\)-axis (if \(A<0\), reflection
+across \(y\)-axis)\\
+\(n\) - dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\),
+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}}
+
+\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}}
+
+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
+
+\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/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}}
+
+\begin{longtable}[]{@{}ll@{}}
+\toprule
+\(n\) is even: & \(n\) is odd:\tabularnewline
+\midrule
+\endhead
+\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/square-root-graph.png}
+&
+\includegraphics[width=0.2\textwidth,height=\textheight]{graphics/cube-root-graph.png}\tabularnewline
+\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}}
+
+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.
+
+\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}}
+
+\[x^{p \over q} = \sqrt[q]{x^p}\]
+
+\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}
+
+\hypertarget{combinations-of-functions-piecewisehybrid}{%
+\subsection{Combinations of functions
+(piecewise/hybrid)}\label{combinations-of-functions-piecewisehybrid}}
+
+\[\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
+
+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}}
+
+\begin{longtable}[]{@{}lll@{}}
+\toprule
+\endhead
+sum & \(f+g\) & domain
+\(= \text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
+difference & \(f-g\) or \(g-f\) & domain
+\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
+product & \(f \times g\) & domain
+\(=\text{dom}(f) \cap \text{dom}(g)\)\tabularnewline
+\bottomrule
+\end{longtable}
+
+Addition of linear piecewise graphs - add \(y\)-values at key points
+
+Product functions:
+
+\begin{itemize}
+\tightlist
+\item
+  product will equal 0 if one of the functions is equal to 0
+\item
+  turning point on one function does not equate to turning point on
+  product
+\end{itemize}
+
+\hypertarget{matrix-transformations}{%
+\subsection{Matrix transformations}\label{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}}
+
+\((f \circ g)(x)\) is defined iff
+\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
+
+\end{document}