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  70\pagestyle{fancy}
  71\fancyhead[LO,LE]{Year 12 Methods}
  72\fancyhead[CO,CE]{Andrew Lorimer}
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 100
 101
 102\begin{document}
 103
 104\title{\vspace{-20mm}Year 12 Methods}
 105\author{Andrew Lorimer}
 106\date{}
 107\maketitle
 108
 109\begin{multicols}{2}
 110
 111
 112\section{Functions}
 113
 114\begin{itemize} \tightlist
 115  \item vertical line test
 116  \item each \(x\) value produces only one \(y\) value
 117\end{itemize}
 118
 119\subsection*{One to one functions}
 120
 121\begin{itemize} \tightlist
 122  \item
 123    \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
 124    \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
 125    \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
 126    \(x^3\) is)
 127  \item
 128    horizontal line test
 129  \item
 130    if not one to one, it is many to one
 131\end{itemize}
 132
 133\subsection*{Odd and even functions}
 134
 135\begin{align*}
 136  \text{Even:}&& f(x)  &= f(-x) \\
 137  \text{Odd:} && -f(x) &= f(-x)
 138\end{align*}
 139
 140Even \(\implies\) symmetrical across \(y\)-axis \\
 141\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
 142For \(x^n\), parity of \(n \equiv\) parity of function
 143
 144\begin{tabularx}{\columnwidth}{XX}
 145  \textbf{Even:} & \textbf{Odd:} \\
 146  \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} &
 147    \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}
 148\end{tabularx}
 149
 150\subsection*{Inverse functions}
 151
 152\begin{itemize} \tightlist
 153  \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
 154  \item \(f\) must be one to one
 155  \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
 156  \item Represents reflection across \(y=x\)
 157  \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
 158  \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
 159    \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
 160  \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
 161\end{itemize}
 162
 163\subsubsection*{Finding \(f^{-1}\)}
 164
 165\begin{enumerate} \tightlist
 166  \item Let \(y=f(x)\)
 167  \item Swap \(x\) and \(y\) (``take inverse''
 168  \item Solve for \(y\) \\
 169    Sqrt: state \(\pm\) solutions then restrict
 170  \item State rule as \(f^{-1}(x)=\dots\)
 171  \item For inverse \emph{function}, state in function notation
 172\end{enumerate}
 173
 174\subsection*{Simultaneous equations (linear)}
 175
 176\begin{itemize} \tightlist
 177  \item \textbf{Unique solution} - lines intersect at point
 178  \item \textbf{Infinitely many solutions} - lines are equal
 179  \item \textbf{No solution} - lines are parallel
 180\end{itemize}
 181
 182\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
 183  where all coefficients are known except for one, and \(a, b\) are known
 184
 185  \begin{enumerate} \tightlist
 186    \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 187      \item Find determinant of first matrix: \(\Delta = ps-qr\)
 188      \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
 189        or let \(\Delta \ne 0\) for one unique solution.
 190      \item Solve determinant equation to find variable \\
 191        \textbf{For infinite/no solutions:}
 192      \item Substitute variable into both original equations
 193      \item Rearrange equations so that LHS of each is the same
 194      \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
 195        \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
 196  \end{enumerate}
 197
 198  \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
 199
 200  \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
 201
 202    \begin{itemize} \tightlist
 203      \item Use elimination
 204      \item Generate two new equations with only two variables
 205      \item Rearrange \& solve
 206      \item Substitute one variable into another equation to find another variable
 207    \end{itemize}
 208
 209    \subsection*{Piecewise functions}
 210
 211    \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
 212
 213      \textbf{Open circle:} point included\\
 214      \textbf{Closed circle:} point not included
 215
 216      \subsection*{Operations on functions}
 217
 218      For \(f \pm g\) and \(f \times g\):
 219      \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
 220
 221      Addition of linear piecewise graphs: add \(y\)-values at key points
 222
 223      Product functions:
 224
 225      \begin{itemize}
 226          \tightlist
 227        \item
 228          product will equal 0 if \(f=0\) or \(g=0\)
 229        \item
 230          \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
 231      \end{itemize}
 232
 233      \subsection*{Composite functions}
 234
 235      \((f \circ g)(x)\) is defined iff
 236      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
 237
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 250      }
 251
 252      \begin{figure*}[ht]
 253        \centering
 254
 255        \begin{tabularx}{\textwidth}{r|Y|Y}
 256
 257          & \(n\) is even & \(n\) is odd \\ \hline
 258
 259          \centering \(x^n, n \in \mathbb{Z}^+\) & 
 260
 261          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 262            \begin{axis}[blank, xmin=-3,  xmax=3]
 263              \addplot[blankplot] {(x^2)};
 264            \end{axis}
 265          \end{tikzpicture}} &
 266
 267          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 268            \begin{axis}[blank, xmin=-3,  xmax=3]
 269              \addplot[blankplot, domain=-3:3] {(x^3)};
 270            \end{axis}
 271          \end{tikzpicture}} \\ \hline
 272
 273          \centering \(x^n, n \in \mathbb{Z}^-\) &
 274
 275          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 276            \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
 277              \addplot[blankplot, samples=100] {(x^(-2))};
 278            \end{axis}
 279          \end{tikzpicture}} &
 280
 281          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 282            \begin{axis}[blank, xmin=-3, xmax=3]
 283              \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
 284              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
 285            \end{axis}
 286          \end{tikzpicture}} \\ \hline
 287
 288          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
 289
 290          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 291            \begin{axis}[blank, xmin=-1,  xmax=5]
 292              \addplot[blankplot] {(x^(1/2))};
 293            \end{axis}
 294          \end{tikzpicture}} &
 295
 296          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
 297            \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
 298              \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
 299            \end{axis}
 300          \end{tikzpicture}} \\ \hline
 301
 302        \end{tabularx}
 303      \end{figure*}
 304
 305      \section{Polynomials}
 306
 307      \subsection*{Linear equations}
 308
 309      \subsubsection*{Forms}
 310
 311      \begin{itemize}
 312          \tightlist
 313        \item \(y=mx+c\)
 314        \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
 315        \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
 316      \end{itemize}
 317
 318      \subsubsection*{Line properties}
 319
 320      Parallel lines: \(m_1 = m_2\)\\
 321      Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 322      Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 323
 324      \subsection*{Quadratics}
 325      \setlength{\abovedisplayskip}{1pt}
 326      \setlength{\belowdisplayskip}{1pt}
 327      \[ x^2 + bx + c = (x+m)(x+n) \]
 328      \hfill where \(mn=c, \> m+n=b\)
 329
 330      \textbf{Difference of squares}
 331      \[ a^2 - b^2 = (a-b)(a+b) \]
 332      \textbf{Perfect squares}
 333      \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
 334      \textbf{Completing the square}
 335      \begin{align*}
 336        x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
 337        ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
 338      \end{align*}
 339      \textbf{Quadratic formula}
 340      \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
 341      \hfill (Discriminant \(\Delta=b^2-4ac\))
 342
 343      \subsection*{Cubics}
 344
 345      \textbf{Difference of cubes}
 346      \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
 347      \textbf{Sum of cubes}
 348      \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
 349      \textbf{Perfect cubes}
 350      \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
 351
 352      \[ y=a(bx-h)^3 + c \]
 353
 354      \begin{itemize}
 355          \tightlist
 356        \item
 357          \(m=0\) at \emph{stationary point of inflection}
 358          (i.e.~(\({h \over b}, k)\))
 359        \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
 360        \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
 361        \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
 362      \end{itemize}
 363
 364      \subsection*{Quartic graphs}
 365
 366      \subsubsection*{Forms of quartic equations}
 367
 368      \(y=ax^4\)\\
 369      \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 370      \(y=ax^4+cd^2 (c \ge 0)\)\\
 371      \(y=ax^2(x-b)(x-c)\)\\
 372      \(y=a(x-b)^2(x-c)^2\)\\
 373      \(y=a(x-b)(x-c)^3\)
 374
 375      \input{transformations}
 376      \input{stuff}
 377      \input{circ-functions}
 378      \input{calculus}
 379
 380      \subfile{statistics-ref}
 381
 382    \end{multicols}
 383
 384\end{document}