b853626a19fc5b180167e9e4869a8a7e859f7a34
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  26\usepackage{longtable}
  27\usepackage{fancyhdr}
  28\pagestyle{fancy}
  29\fancyhead[LO,LE]{Year 12 Methods}
  30\fancyhead[CO,CE]{Andrew Lorimer}
  31\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
  32\setlength{\parindent}{0cm}
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  57\begin{document}
  58
  59\title{\vspace{-2cm}\hrule\vspace{0.4cm} Year 12 Methods}
  60\author{Andrew Lorimer}
  61\date{}
  62\maketitle
  63
  64\begin{multicols}{2}
  65
  66\section{Functions}
  67
  68\begin{itemize}
  69  \tightlist
  70  \item vertical line test
  71  \item each \(x\) value produces only one \(y\) value
  72\end{itemize}
  73
  74\subsection*{One to one functions}
  75
  76\begin{itemize}
  77\tightlist
  78\item
  79  \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
  80  \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
  81  \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
  82  \(x^3\) is)
  83\item
  84  horizontal line test
  85\item
  86  if not one to one, it is many to one
  87\end{itemize}
  88
  89\subsection*{Finding inverse functions \(f^{-1}\)}
  90
  91\begin{itemize}
  92\tightlist
  93\item
  94  if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
  95\item
  96  reflection across \(y-x\)
  97\item
  98  \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
  99\item
 100  inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass
 101  vertical line test)\\
 102  \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one
 103\item
 104  \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\)
 105\end{itemize}
 106
 107\subsubsection*{Requirements for showing working for \(f^{-1}\)}
 108
 109\begin{enumerate}
 110\def\labelenumi{\arabic{enumi}.}
 111\tightlist
 112\item
 113  start with \emph{``let \(y=f(x)\)''}
 114\item
 115  must state \emph{``take inverse''} for line where \(y\) and \(x\) are
 116  swapped
 117\item
 118  do all working in terms of \(y=\dots\)
 119\item
 120  for sqrt, state \(\pm\) solutions then show restricted
 121\item
 122  for inverse \emph{function}, state in function notation
 123\end{enumerate}
 124\subsubsection*{Solving
 125\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
 126for \(\{0,1,\infty\}\)
 127solutions}
 128
 129where all coefficients are known except for one, and \(a, b\) are known
 130
 131\begin{enumerate}
 132\tightlist
 133\item
 134  Write as matrices:
 135  \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 136\item
 137  Find determinant of first matrix: \(\Delta = ps-qr\)
 138\item
 139  Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
 140  or let \(\Delta \ne 0\) for one unique solution.
 141\item
 142  Solve determinant equation to find variable \\
 143    \textbf{For infinite/no solutions:}
 144\item
 145  Substitute variable into both original equations
 146\item
 147  Rearrange equations so that LHS of each is the same
 148\item
 149  \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\)
 150  (\(\infty\) solns)\\
 151  \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0
 152  solns)
 153\end{enumerate}
 154
 155\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
 156
 157\subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
 158
 159\begin{itemize}
 160\tightlist
 161\item
 162  Use elimination
 163\item
 164  Generate two new equations with only two variables
 165\item
 166  Rearrange \& solve
 167\item
 168  Substitute one variable into another equation to find another variable
 169\end{itemize}
 170\subsection*{Odd and even functions}
 171
 172Even when \(f(x) = -f(x)\)\\
 173Odd when \(-f(x) = f(-x)\)
 174
 175Function is even if it is symmetrical across \(y\)-axis
 176\hspace{5em}\(\implies f(x)=f(-x)\)\\
 177Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
 178
 179\begin{tabularx}{\columnwidth}{XX}
 180  \textbf{Even:} & \textbf{Odd:} \\
 181  \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} &
 182  \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}
 183\end{tabularx}
 184\pagebreak
 185                  \pgfplotsset{every axis/.append style={
 186                    xlabel=,    % put the x axis in the middle
 187                    ylabel=,    % put the y axis in the middle
 188                  }}
 189                  \begin{table*}[ht]
 190                    \centering
 191                    \begin{tabularx}{\textwidth}{r|X|X}
 192                      & \(n\) is even & \(n\) is odd \\ \hline
 193                      \(x^n, n \in \mathbb{Z}^+\) & 
 194                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^2)};  \end{axis}\end{tikzpicture}} &
 195                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^3)};  \end{axis}\end{tikzpicture}} \\
 196                      \(x^n, n \in \mathbb{Z}^-\) &
 197                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-4,  xmax=4, ymax=8, ymin=-0, scale=0.4, smooth] \addplot[orange, mark=none, samples=100] {(x^(-2))};  \end{axis}\end{tikzpicture}} &
 198                        \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3,  xmax=3, scale=0.4, samples=100, smooth] \addplot[orange, mark=none] {(x^(-1))};  \end{axis}\end{tikzpicture}} \\
 199                      \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
 200                      \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-1,  xmax=5, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none] {(x^(1/2))};  \end{axis}\end{tikzpicture}} &
 201                        \makecell{\\\begin{tikzpicture}
 202                      \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
 203\addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
 204\end{axis}
 205                        \end{tikzpicture}}
 206                    \end{tabularx}
 207                  \end{table*}
 208                  \pgfplotsset{every axis/.append style={
 209                    xlabel=\(x\),    % put the x axis in the middle
 210                    ylabel=\(y\),    % put the y axis in the middle
 211                  }}
 212
 213\section{Polynomials}
 214
 215\subsection*{Quadratics}
 216
 217\[ x^2 + bx + c = (x+m)(x+n) \]
 218\hfill where \(mn=c, \> m+n=b\)
 219
 220\begin{align*}
 221  \hline
 222  \textbf{Difference} && a^2 - b^2 &= (a-b)(a+b) \\[2ex]
 223  \textbf{Perfect sq.} && a^2 \pm 2ab + b^2 &= (a \pm b^2) \\[2ex]
 224  \textbf{Completing} && x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
 225  && ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a} \\[2ex]
 226  \textbf{Quadratic} && x &= \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \\
 227  && & \text{where} \Delta=b^2-4ac \\
 228  \hline
 229\end{align*}
 230
 231\subsection*{Cubics}
 232
 233\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
 234\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
 235\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
 236
 237\[ y=a(bx-h)^3 + c \]
 238
 239\begin{itemize}
 240\tightlist
 241\item
 242  \(m=0\) at \emph{stationary point of inflection}
 243  (i.e.~(\({h \over b}, k)\))
 244\item
 245  in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
 246\item
 247  in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
 248\item
 249  in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
 250  \(c\)
 251\end{itemize}
 252
 253\subsection*{Linear and quadratic
 254graphs}
 255
 256\subsubsection*{Forms of linear
 257equations}
 258
 259\begin{itemize}
 260\tightlist
 261  \item \(y=mx+c\)
 262  \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
 263  \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
 264\end{itemize}
 265
 266\subsection*{Line properties}
 267
 268Parallel lines: \(m_1 = m_2\)\\
 269Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 270Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 271
 272\subsection*{Quartic graphs}
 273
 274\subsubsection*{Forms of quartic
 275equations}
 276
 277\(y=ax^4\)\\
 278\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 279\(y=ax^4+cd^2 (c \ge 0)\)\\
 280\(y=ax^2(x-b)(x-c)\)\\
 281\(y=a(x-b)^2(x-c)^2\)\\
 282\(y=a(x-b)(x-c)^3\)
 283
 284\subsection*{Simultaneous equations
 285(linear)}
 286
 287\begin{itemize}
 288\tightlist
 289\item
 290  \textbf{Unique solution} - lines intersect at point
 291\item
 292  \textbf{Infinitely many solutions} - lines are equal
 293\item
 294  \textbf{No solution} - lines are parallel
 295\end{itemize}
 296
 297
 298\input{temp/transformations}
 299\input{temp/stuff}
 300\input{circ-functions}
 301\input{temp/calculus}
 302
 303\end{multicols}
 304\end{document}