From bd487b9356352199ce02b9c512ce5352379dff92 Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Sun, 3 Mar 2019 19:16:31 +1100 Subject: [PATCH] add transformations-ref source --- methods/transformations-ref.tex | 288 ++++++++++++++++++++++++++++++++ 1 file changed, 288 insertions(+) create mode 100644 methods/transformations-ref.tex diff --git a/methods/transformations-ref.tex b/methods/transformations-ref.tex new file mode 100644 index 0000000..22532c4 --- /dev/null +++ b/methods/transformations-ref.tex @@ -0,0 +1,288 @@ +\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} -- 2.43.2