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  39  pdfauthor={Andrew Lorimer},
  40  hidelinks,
  41  pdfcreator={LaTeX via pandoc}}
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  51\fancyhead[LO,LE]{Year 12 Methods}
  52\fancyhead[CO,CE]{Andrew Lorimer}
  53\usepackage{graphicx}
  54\usepackage{tabularx}
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  56
  57\author{Andrew Lorimer}
  58\date{}
  59
  60\begin{document}
  61
  62\hypertarget{polynomials}{%
  63\section{Polynomials}\label{polynomials}}
  64
  65\hypertarget{quadratics}{%
  66\subsection{Quadratics}\label{quadratics}}
  67
  68\newcolumntype{R}{>{\raggedleft\arraybackslash}X}
  69\begin{tabularx}{\columnwidth}{Rl}
  70  General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\
  71  \hline
  72  Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\
  73  \hline
  74  Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
  75  \hline
  76  Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
  77  \hline
  78  Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
  79\end{tabularx}
  80
  81\hypertarget{cubics}{%
  82\subsection{Cubics}\label{cubics}}
  83
  84\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\
  85\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\
  86\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\)
  87
  88\[y=a(bx-h)^3 + c\]
  89
  90\begin{itemize}
  91\tightlist
  92\item
  93  \(m=0\) at \emph{stationary point of inflection}
  94  (i.e.~(\({h \over b}, k)\))
  95\item
  96  in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\)
  97\item
  98  in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\)
  99\item
 100  in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at
 101  \(c\)
 102\end{itemize}
 103
 104\hypertarget{linear-and-quadratic-graphs}{%
 105\subsection{Linear and quadratic
 106graphs}\label{linear-and-quadratic-graphs}}
 107
 108\hypertarget{forms-of-linear-equations}{%
 109\subsubsection{Forms of linear
 110equations}\label{forms-of-linear-equations}}
 111
 112\(y=mx+c\) where \(m\) is gradient and \(c\) is \(y\)-intercept\\
 113\({x \over a} + {y \over b}=1\) where \(m\) is gradient and
 114\((x_1, y_1)\) lies on the graph\\
 115\(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and
 116\(y\)-intercepts
 117
 118\hypertarget{line-properties}{%
 119\subsection{Line properties}\label{line-properties}}
 120
 121Parallel lines: \(m_1 = m_2\)\\
 122Perpendicular lines: \(m_1 \times m_2 = -1\)\\
 123Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
 124
 125\hypertarget{quartic-graphs}{%
 126\subsection{Quartic graphs}\label{quartic-graphs}}
 127
 128\hypertarget{forms-of-quadratic-equations}{%
 129\subsubsection{Forms of quadratic
 130equations}\label{forms-of-quadratic-equations}}
 131
 132\(y=ax^4\)\\
 133\(y=a(x-b)(x-c)(x-d)(x-e)\)\\
 134\(y=ax^4+cd^2 (c \ge 0)\)\\
 135\(y=ax^2(x-b)(x-c)\)\\
 136\(y=a(x-b)^2(x-c)^2\)\\
 137\(y=a(x-b)(x-c)^3\)
 138
 139\hypertarget{simultaneous-equations-linear}{%
 140\subsection{Simultaneous equations
 141(linear)}\label{simultaneous-equations-linear}}
 142
 143\begin{itemize}
 144\tightlist
 145\item
 146  \textbf{Unique solution} - lines intersect at point
 147\item
 148  \textbf{Infinitely many solutions} - lines are equal
 149\item
 150  \textbf{No solution} - lines are parallel
 151\end{itemize}
 152
 153\hypertarget{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}{%
 154\subsubsection{\texorpdfstring{Solving
 155\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\)
 156for \(\{0,1,\infty\}\)
 157solutions}{Solving \textbackslash protect\textbackslash begin\{cases\}px + qy = a \textbackslash\textbackslash{} rx + sy = b\textbackslash protect\textbackslash end\{cases\} \textbackslash\textgreater{} for \textbackslash\{0,1,\textbackslash infty\textbackslash\} solutions}}\label{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}}
 158
 159where all coefficients are known except for one, and \(a, b\) are known
 160
 161\begin{enumerate}
 162\def\labelenumi{\arabic{enumi}.}
 163\tightlist
 164\item
 165  Write as matrices:
 166  \(\begin{bmatrix}p & q \\ r & s \end{bmatrix}  \begin{bmatrix} x \\ y \end{bmatrix}  =  \begin{bmatrix} a \\ b \end{bmatrix}\)
 167\item
 168  Find determinant of first matrix: \(\Delta = ps-qr\)
 169\item
 170  Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
 171  or let \(\Delta \ne 0\) for one unique solution.
 172\item
 173  Solve determinant equation to find variable
 174
 175  \begin{itemize}
 176  \tightlist
 177  \item
 178    \emph{--- for infinite/no solutions: ---}
 179  \end{itemize}
 180\item
 181  Substitute variable into both original equations
 182\item
 183  Rearrange equations so that LHS of each is the same
 184\item
 185  \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\)
 186  (\(\infty\) solns)\\
 187  \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0
 188  solns)
 189\end{enumerate}
 190
 191\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
 192
 193\hypertarget{solving-protectbegincasesa_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_3protectendcases}{%
 194\subsubsection{\texorpdfstring{Solving
 195\(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}{Solving \textbackslash protect\textbackslash begin\{cases\}a\_1 x + b\_1 y + c\_1 z = d\_1 \textbackslash\textbackslash{} a\_2 x + b\_2 y + c\_2 z = d\_2 \textbackslash\textbackslash{} a\_3 x + b\_3 y + c\_3 z = d\_3\textbackslash protect\textbackslash end\{cases\}}}\label{solving-protectbegincasesa_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_3protectendcases}}
 196
 197\begin{itemize}
 198\tightlist
 199\item
 200  Use elimination
 201\item
 202  Generate two new equations with only two variables
 203\item
 204  Rearrange \& solve
 205\item
 206  Substitute one variable into another equation to find another variable
 207\item
 208  etc.
 209\end{itemize}
 210
 211\end{document}