b853626a19fc5b180167e9e4869a8a7e859f7a34
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5\usepackage{multicol}
6\usepackage{multirow}
7\usepackage{amsmath}
8\usepackage{amssymb}
9\usepackage{harpoon}
10\usepackage{tabularx}
11\usepackage{makecell}
12\usepackage[dvipsnames, table]{xcolor}
13\usepackage{blindtext}
14\usepackage{graphicx}
15\usepackage{wrapfig}
16\usepackage{tikz}
17\usepackage{tikz-3dplot}
18\usepackage{pgfplots}
19\pgfplotsset{compat=1.8}
20\usepackage{mathtools}
21\usetikzlibrary{calc}
22\usetikzlibrary{angles}
23\usetikzlibrary{datavisualization.formats.functions}
24\usetikzlibrary{decorations.markings}
25\usepgflibrary{arrows.meta}
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}
33\usepackage{mathtools}
34\usepackage{xcolor} % used only to show the phantomed stuff
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56}}
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}