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27\usepackage{longtable}
28\usepackage{fancyhdr}
29\pagestyle{fancy}
30\fancyhead[LO,LE]{Year 12 Methods}
31\fancyhead[CO,CE]{Andrew Lorimer}
32\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
33\providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}}
34\setlength{\parindent}{0cm}
35\usepackage{mathtools}
36\usepackage{xcolor} % used only to show the phantomed stuff
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59
60\begin{document}
61
62\title{\vspace{-20mm}Year 12 Methods}
63\author{Andrew Lorimer}
64\date{}
65\maketitle
66
67\begin{multicols}{2}
68
69 \section{Functions}
70
71 \begin{itemize}
72 \tightlist
73 \item vertical line test
74 \item each \(x\) value produces only one \(y\) value
75 \end{itemize}
76
77 \subsection*{One to one functions}
78
79 \begin{itemize} \tightlist
80 \item
81 \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
82 \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
83 \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
84 \(x^3\) is)
85 \item
86 horizontal line test
87 \item
88 if not one to one, it is many to one
89 \end{itemize}
90
91 \subsection*{Odd and even functions}
92
93 \begin{align*}
94 \text{Even:}&& f(x) &= f(-x) \\
95 \text{Odd:} && -f(x) &= f(-x)
96 \end{align*}
97
98 Even \(\implies\) symmetrical across \(y\)-axis \\
99 \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
100 For \(x^n\), parity of \(n \equiv\) parity of function
101
102 \begin{tabularx}{\columnwidth}{XX}
103 \textbf{Even:} & \textbf{Odd:} \\
104 \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} &
105 \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}
106 \end{tabularx}
107
108 \subsection*{Inverse functions}
109
110 \begin{itemize} \tightlist
111 \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
112 \item \(f\) must be one to one
113 \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
114 \item Represents reflection across \(y=x\)
115 \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
116 \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
117 \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
118 \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
119 \end{itemize}
120
121 \subsubsection*{Finding \(f^{-1}\)}
122
123 \begin{enumerate} \tightlist
124 \item Let \(y=f(x)\)
125 \item Swap \(x\) and \(y\) (``take inverse''
126 \item Solve for \(y\) \\
127 Sqrt: state \(\pm\) solutions then restrict
128 \item State rule as \(f^{-1}(x)=\dots\)
129 \item For inverse \emph{function}, state in function notation
130 \end{enumerate}
131
132 \subsection*{Simultaneous equations (linear)}
133
134 \begin{itemize} \tightlist
135 \item \textbf{Unique solution} - lines intersect at point
136 \item \textbf{Infinitely many solutions} - lines are equal
137 \item \textbf{No solution} - lines are parallel
138 \end{itemize}
139
140 \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
141 where all coefficients are known except for one, and \(a, b\) are known
142
143 \begin{enumerate} \tightlist
144 \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
145 \item Find determinant of first matrix: \(\Delta = ps-qr\)
146 \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
147 or let \(\Delta \ne 0\) for one unique solution.
148 \item Solve determinant equation to find variable \\
149 \textbf{For infinite/no solutions:}
150 \item Substitute variable into both original equations
151 \item Rearrange equations so that LHS of each is the same
152 \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
153 \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
154 \end{enumerate}
155
156 \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
157
158 \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
159
160 \begin{itemize} \tightlist
161 \item Use elimination
162 \item Generate two new equations with only two variables
163 \item Rearrange \& solve
164 \item Substitute one variable into another equation to find another variable
165 \end{itemize}
166
167\subsection*{Piecewise functions}
168
169\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
170
171\textbf{Open circle:} point included\\
172\textbf{Closed circle:} point not included
173
174\subsection*{Operations on functions}
175
176For \(f \pm g\) and \(f \times g\):
177\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
178
179Addition of linear piecewise graphs: add \(y\)-values at key points
180
181Product functions:
182
183\begin{itemize}
184\tightlist
185\item
186 product will equal 0 if \(f=0\) or \(g=0\)
187\item
188 \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
189\end{itemize}
190
191\subsection*{Composite functions}
192
193\((f \circ g)(x)\) is defined iff
194\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
195
196
197 \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks
198 \begin{table*}[ht]
199 \centering
200 \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}}
201 & \(n\) is even & \(n\) is odd \\ \tabucline{1pt}
202 \(x^n, n \in \mathbb{Z}^+\) &
203 \vspace{1em}\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} &
204 \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} \\
205 \(x^n, n \in \mathbb{Z}^-\) &
206 \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} &
207 \begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth, unbounded coords=jump] \addplot[orange, mark=none, domain=-3:-0.1] {(x^(-1))}; \addplot[orange, mark=none, domain=0.1:3] {(x^(-1))}; \end{axis}\end{tikzpicture} \\
208 \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
209 \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} &
210 \begin{tikzpicture}
211 \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4]
212 \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
213 \end{axis}
214 \end{tikzpicture}
215 \end{tabu}
216 \hrule
217 \end{table*}
218 \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back
219
220 \section{Polynomials}
221
222 \subsection*{Linear equations}
223
224 \subsubsection*{Forms}
225
226 \begin{itemize}
227 \tightlist
228 \item \(y=mx+c\)
229 \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
230 \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
231 \end{itemize}
232
233 \subsubsection*{Line properties}
234
235 Parallel lines: \(m_1 = m_2\)\\
236 Perpendicular lines: \(m_1 \times m_2 = -1\)\\
237 Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
238
239 \subsection*{Quadratics}
240 \setlength{\abovedisplayskip}{1pt}
241 \setlength{\belowdisplayskip}{1pt}
242 \[ x^2 + bx + c = (x+m)(x+n) \]
243 \hfill where \(mn=c, \> m+n=b\)
244
245 \textbf{Difference of squares}
246 \[ a^2 - b^2 = (a-b)(a+b) \]
247 \textbf{Perfect squares}
248 \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
249 \textbf{Completing the square}
250 \begin{align*}
251 x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
252 ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
253 \end{align*}
254 \textbf{Quadratic formula}
255 \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
256 \hfill (Discriminant \(\Delta=b^2-4ac\))
257
258 \subsection*{Cubics}
259
260 \textbf{Difference of cubes}
261 \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
262 \textbf{Sum of cubes}
263 \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
264 \textbf{Perfect cubes}
265 \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
266
267 \[ y=a(bx-h)^3 + c \]
268
269 \begin{itemize}
270 \tightlist
271 \item
272 \(m=0\) at \emph{stationary point of inflection}
273 (i.e.~(\({h \over b}, k)\))
274 \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
275 \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
276 \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
277 \end{itemize}
278
279 \subsection*{Quartic graphs}
280
281 \subsubsection*{Forms of quartic equations}
282
283 \(y=ax^4\)\\
284 \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
285 \(y=ax^4+cd^2 (c \ge 0)\)\\
286 \(y=ax^2(x-b)(x-c)\)\\
287 \(y=a(x-b)^2(x-c)^2\)\\
288 \(y=a(x-b)(x-c)^3\)
289
290 \input{transformations}
291 \input{stuff}
292 \input{circ-functions}
293 \input{calculus}
294
295 \end{multicols}
296 \end{document}