b0641308c27af5346bcda07e675f0124b9def6ef
   1\documentclass[a4paper]{article}
   2\usepackage[dvipsnames, table]{xcolor}
   3\usepackage{adjustbox}
   4\usepackage{amsmath}
   5\usepackage{amssymb}
   6\usepackage{blindtext}
   7\usepackage{enumitem}
   8\usepackage{fancyhdr}
   9\usepackage[a4paper,margin=2cm]{geometry}
  10\usepackage{graphicx}
  11\usepackage{harpoon}
  12\usepackage{listings}
  13\usepackage{longtable}
  14\usepackage{makecell}
  15\usepackage{mathtools}
  16\usepackage{multicol}
  17\usepackage{multirow}
  18\usepackage{newclude}
  19\usepackage{pgfplots}
  20\usepackage{pst-plot}
  21\usepackage{standalone}
  22\usepackage{subfiles}
  23\usepackage{tabularx}
  24\usepackage{tabu}
  25\usepackage{tcolorbox}
  26\usepackage{tikz-3dplot}
  27\usepackage{tikz}
  28\usepackage{tkz-fct}
  29\usepackage[obeyspaces]{url}
  30\usepackage{wrapfig}
  31
  32
  33\usetikzlibrary{%
  34  angles,
  35  calc,
  36  datavisualization.formats.functions,
  37  decorations,
  38  decorations.markings,
  39  decorations.pathreplacing,
  40  decorations.text,
  41  scopes
  42}
  43\newcommand{\midarrow}{\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}
  44\usepgflibrary{arrows.meta}
  45\pgfplotsset{compat=1.6}
  46\psset{dimen=monkey,fillstyle=solid,opacity=.5}
  47\def\object{%
  48    \psframe[linestyle=none,fillcolor=blue](-2,-1)(2,1)
  49    \psaxes[linecolor=gray,labels=none,ticks=none]{->}(0,0)(-3,-3)(3,2)[$x$,0][$y$,90]
  50    \rput{*0}{%
  51        \psline{->}(0,-2)%
  52        \uput[-90]{*0}(0,-2){$\vec{w}$}}
  53}
  54\newcommand{\tg}{\mathop{\mathrm{tg}}}
  55\newcommand{\cotg}{\mathop{\mathrm{cotg}}}
  56\newcommand{\arctg}{\mathop{\mathrm{arctg}}}
  57\newcommand{\arccotg}{\mathop{\mathrm{arccotg}}}
  58\pgfplotsset{every axis/.append style={
  59  axis x line=middle,    % centre axes
  60  axis y line=middle,
  61  axis line style={->},  % arrows on axes
  62  xlabel={$x$},          % axes labels
  63  ylabel={$y$}
  64}}
  65
  66\pagestyle{fancy}
  67\fancyhead[LO,LE]{Year 12 Methods}
  68\fancyhead[CO,CE]{Andrew Lorimer}
  69\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
  70
  71\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
  72\linespread{1.5}
  73\setlength{\parindent}{0cm}
  74\setlength\fboxsep{0pt} \setlength\fboxrule{.2pt} % for the \fboxes
  75\newcommand*\leftlap[3][\,]{#1\hphantom{#2}\mathllap{#3}}
  76\newcommand*\rightlap[2]{\mathrlap{#2}\hphantom{#1}}
  77
  78\newcolumntype{L}[1]{>{\hsize=#1\hsize\raggedright\arraybackslash}X}
  79\newcolumntype{R}[1]{>{\hsize=#1\hsize\raggedleft\arraybackslash}X}
  80\newcolumntype{Y}{>{\centering\arraybackslash}X}
  81
  82\definecolor{cas}{HTML}{e6f0fe}
  83\definecolor{important}{HTML}{fc9871}
  84\definecolor{dark-gray}{gray}{0.2}
  85\definecolor{shade1}{HTML}{ffffff}
  86\definecolor{shade2}{HTML}{e6f2ff}
  87\definecolor{shade3}{HTML}{cce2ff}
  88
  89\newtcolorbox{cas}{colframe=cas!75!black, title=On CAS, left*=3mm}
  90\newtcolorbox{warning}{colback=white!90!black, leftrule=3mm, colframe=important, coltext=important, fontupper=\sffamily\bfseries}
  91
  92
  93\begin{document}
  94
  95\title{\vspace{-20mm}Year 12 Methods}
  96\author{Andrew Lorimer}
  97\date{}
  98\maketitle
  99
 100\begin{multicols}{2}
 101
 102
 103\section{Functions}
 104
 105\begin{itemize} \tightlist
 106  \item vertical line test
 107  \item each \(x\) value produces only one \(y\) value
 108\end{itemize}
 109
 110\subsection*{One to one functions}
 111
 112\begin{itemize} \tightlist
 113  \item
 114    \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
 115    \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
 116    \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
 117    \(x^3\) is)
 118  \item
 119    horizontal line test
 120  \item
 121    if not one to one, it is many to one
 122\end{itemize}
 123
 124\subsection*{Odd and even functions}
 125
 126\begin{align*}
 127  \text{Even:}&& f(x)  &= f(-x) \\
 128  \text{Odd:} && -f(x) &= f(-x)
 129\end{align*}
 130
 131Even \(\implies\) symmetrical across \(y\)-axis \\
 132\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
 133For \(x^n\), parity of \(n \equiv\) parity of function
 134
 135\begin{tabularx}{\columnwidth}{XX}
 136  \textbf{Even:} & \textbf{Odd:} \\
 137  \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^2)};  \end{axis}\end{tikzpicture} &
 138    \begin{tikzpicture}\begin{axis}[ticks=none, yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[blue, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}
 139\end{tabularx}
 140
 141\subsection*{Inverse functions}
 142
 143\begin{itemize} \tightlist
 144  \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
 145  \item \(f\) must be one to one
 146  \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
 147  \item Represents reflection across \(y=x\)
 148  \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
 149  \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
 150    \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
 151  \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
 152\end{itemize}
 153
 154\subsubsection*{Finding \(f^{-1}\)}
 155
 156\begin{enumerate} \tightlist
 157  \item Let \(y=f(x)\)
 158  \item Swap \(x\) and \(y\) (``take inverse''
 159  \item Solve for \(y\) \\
 160    Sqrt: state \(\pm\) solutions then restrict
 161  \item State rule as \(f^{-1}(x)=\dots\)
 162  \item For inverse \emph{function}, state in function notation
 163\end{enumerate}
 164
 165\subsection*{Simultaneous equations (linear)}
 166
 167\begin{itemize} \tightlist
 168  \item \textbf{Unique solution} - lines intersect at point
 169  \item \textbf{Infinitely many solutions} - lines are equal
 170  \item \textbf{No solution} - lines are parallel
 171\end{itemize}
 172
 173\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
 174  where all coefficients are known except for one, and \(a, b\) are known
 175
 176  \begin{enumerate} \tightlist
 177    \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 178      \item Find determinant of first matrix: \(\Delta = ps-qr\)
 179      \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
 180        or let \(\Delta \ne 0\) for one unique solution.
 181      \item Solve determinant equation to find variable \\
 182        \textbf{For infinite/no solutions:}
 183      \item Substitute variable into both original equations
 184      \item Rearrange equations so that LHS of each is the same
 185      \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
 186        \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
 187  \end{enumerate}
 188
 189  \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
 190
 191  \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
 192
 193    \begin{itemize} \tightlist
 194      \item Use elimination
 195      \item Generate two new equations with only two variables
 196      \item Rearrange \& solve
 197      \item Substitute one variable into another equation to find another variable
 198    \end{itemize}
 199
 200    \subsection*{Piecewise functions}
 201
 202    \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 203
 204      \textbf{Open circle:} point included\\
 205      \textbf{Closed circle:} point not included
 206
 207      \subsection*{Operations on functions}
 208
 209      For \(f \pm g\) and \(f \times g\):
 210      \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
 211
 212      Addition of linear piecewise graphs: add \(y\)-values at key points
 213
 214      Product functions:
 215
 216      \begin{itemize}
 217          \tightlist
 218        \item
 219          product will equal 0 if \(f=0\) or \(g=0\)
 220        \item
 221          \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
 222      \end{itemize}
 223
 224      \subsection*{Composite functions}
 225
 226      \((f \circ g)(x)\) is defined iff
 227      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 228
 229      \pgfplotsset{
 230        blank/.append style={%
 231          enlargelimits=true,
 232          ticks=none,
 233          yticklabels={,,}, xticklabels={,,},
 234          xlabel=, ylabel=,
 235          scale=0.4,
 236          samples=100, smooth, unbounded coords=jump
 237        }
 238      }
 239      \tikzset{
 240        blankplot/.append style={orange, mark=none}
 241      }
 242
 243      \begin{figure*}[ht]
 244        \centering
 245
 246        \begin{tabularx}{\textwidth}{r|Y|Y}
 247
 248          & \(n\) is even & \(n\) is odd \\ \hline
 249
 250          \centering \(x^n, n \in \mathbb{Z}^+\) & 
 251
 252          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 253            \begin{axis}[blank, xmin=-3,  xmax=3]
 254              \addplot[blankplot] {(x^2)};
 255            \end{axis}
 256          \end{tikzpicture}} &
 257
 258          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 259            \begin{axis}[blank, xmin=-3,  xmax=3]
 260              \addplot[blankplot, domain=-3:3] {(x^3)};
 261            \end{axis}
 262          \end{tikzpicture}} \\ \hline
 263
 264          \centering \(x^n, n \in \mathbb{Z}^-\) &
 265
 266          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 267            \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
 268              \addplot[blankplot, samples=100] {(x^(-2))};
 269            \end{axis}
 270          \end{tikzpicture}} &
 271
 272          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 273            \begin{axis}[blank, xmin=-3, xmax=3]
 274              \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
 275              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
 276            \end{axis}
 277          \end{tikzpicture}} \\ \hline
 278
 279          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
 280
 281          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 282            \begin{axis}[blank, xmin=-1,  xmax=5]
 283              \addplot[blankplot] {(x^(1/2))};
 284            \end{axis}
 285          \end{tikzpicture}} &
 286
 287          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 288            \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
 289              \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
 290            \end{axis}
 291          \end{tikzpicture}} \\ \hline
 292
 293        \end{tabularx}
 294      \end{figure*}
 295
 296      \section{Polynomials}
 297
 298      \subsection*{Linear equations}
 299
 300      \subsubsection*{Forms}
 301
 302      \begin{itemize}
 303          \tightlist
 304        \item \(y=mx+c\)
 305        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
 306        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
 307      \end{itemize}
 308
 309      \subsubsection*{Line properties}
 310
 311      Parallel lines: \(m_1 = m_2\)\\
 312      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 313      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 314
 315      \subsection*{Quadratics}
 316      \setlength{\abovedisplayskip}{1pt}
 317      \setlength{\belowdisplayskip}{1pt}
 318      \[ x^2 + bx + c = (x+m)(x+n) \]
 319      \hfill where \(mn=c, \> m+n=b\)
 320
 321      \textbf{Difference of squares}
 322      \[ a^2 - b^2 = (a-b)(a+b) \]
 323      \textbf{Perfect squares}
 324      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
 325      \textbf{Completing the square}
 326      \begin{align*}
 327        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
 328        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
 329      \end{align*}
 330      \textbf{Quadratic formula}
 331      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
 332      \hfill (Discriminant \(\Delta=b^2-4ac\))
 333
 334      \subsection*{Cubics}
 335
 336      \textbf{Difference of cubes}
 337      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
 338      \textbf{Sum of cubes}
 339      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
 340      \textbf{Perfect cubes}
 341      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
 342
 343      \[ y=a(bx-h)^3 + c \]
 344
 345      \begin{itemize}
 346          \tightlist
 347        \item
 348          \(m=0\) at \emph{stationary point of inflection}
 349          (i.e.~(\({h \over b}, k)\))
 350        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
 351        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
 352        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
 353      \end{itemize}
 354
 355      \subsection*{Quartic graphs}
 356
 357      \subsubsection*{Forms of quartic equations}
 358
 359      \(y=ax^4\)\\
 360      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 361      \(y=ax^4+cd^2 (c \ge 0)\)\\
 362      \(y=ax^2(x-b)(x-c)\)\\
 363      \(y=a(x-b)^2(x-c)^2\)\\
 364      \(y=a(x-b)(x-c)^3\)
 365
 366      \input{transformations}
 367      \input{stuff}
 368      \input{circ-functions}
 369      \input{calculus}
 370
 371      \subfile{statistics-ref}
 372
 373    \end{multicols}
 374
 375\end{document}