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66\pagestyle{fancy}
67\fancyhead[LO,LE]{Year 12 Methods}
68\fancyhead[CO,CE]{Andrew Lorimer}
69\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
70
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91
92
93\begin{document}
94
95\title{\vspace{-20mm}Year 12 Methods}
96\author{Andrew Lorimer}
97\date{}
98\maketitle
99
100\begin{multicols}{2}
101
102
103\section{Functions}
104
105\begin{itemize} \tightlist
106 \item vertical line test
107 \item each \(x\) value produces only one \(y\) value
108\end{itemize}
109
110\subsection*{One to one functions}
111
112\begin{itemize} \tightlist
113 \item
114 \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
115 \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
116 \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
117 \(x^3\) is)
118 \item
119 horizontal line test
120 \item
121 if not one to one, it is many to one
122\end{itemize}
123
124\subsection*{Odd and even functions}
125
126\begin{align*}
127 \text{Even:}&& f(x) &= f(-x) \\
128 \text{Odd:} && -f(x) &= f(-x)
129\end{align*}
130
131Even \(\implies\) symmetrical across \(y\)-axis \\
132\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
133For \(x^n\), parity of \(n \equiv\) parity of function
134
135\begin{tabularx}{\columnwidth}{XX}
136 \textbf{Even:} & \textbf{Odd:} \\
137 \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} &
138 \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}
139\end{tabularx}
140
141\subsection*{Inverse functions}
142
143\begin{itemize} \tightlist
144 \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
145 \item \(f\) must be one to one
146 \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
147 \item Represents reflection across \(y=x\)
148 \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
149 \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
150 \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
151 \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
152\end{itemize}
153
154\subsubsection*{Finding \(f^{-1}\)}
155
156\begin{enumerate} \tightlist
157 \item Let \(y=f(x)\)
158 \item Swap \(x\) and \(y\) (``take inverse''
159 \item Solve for \(y\) \\
160 Sqrt: state \(\pm\) solutions then restrict
161 \item State rule as \(f^{-1}(x)=\dots\)
162 \item For inverse \emph{function}, state in function notation
163\end{enumerate}
164
165\subsection*{Simultaneous equations (linear)}
166
167\begin{itemize} \tightlist
168 \item \textbf{Unique solution} - lines intersect at point
169 \item \textbf{Infinitely many solutions} - lines are equal
170 \item \textbf{No solution} - lines are parallel
171\end{itemize}
172
173\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
174 where all coefficients are known except for one, and \(a, b\) are known
175
176 \begin{enumerate} \tightlist
177 \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
178 \item Find determinant of first matrix: \(\Delta = ps-qr\)
179 \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
180 or let \(\Delta \ne 0\) for one unique solution.
181 \item Solve determinant equation to find variable \\
182 \textbf{For infinite/no solutions:}
183 \item Substitute variable into both original equations
184 \item Rearrange equations so that LHS of each is the same
185 \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
186 \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
187 \end{enumerate}
188
189 \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
190
191 \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
192
193 \begin{itemize} \tightlist
194 \item Use elimination
195 \item Generate two new equations with only two variables
196 \item Rearrange \& solve
197 \item Substitute one variable into another equation to find another variable
198 \end{itemize}
199
200 \subsection*{Piecewise functions}
201
202 \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
203
204 \textbf{Open circle:} point included\\
205 \textbf{Closed circle:} point not included
206
207 \subsection*{Operations on functions}
208
209 For \(f \pm g\) and \(f \times g\):
210 \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
211
212 Addition of linear piecewise graphs: add \(y\)-values at key points
213
214 Product functions:
215
216 \begin{itemize}
217 \tightlist
218 \item
219 product will equal 0 if \(f=0\) or \(g=0\)
220 \item
221 \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
222 \end{itemize}
223
224 \subsection*{Composite functions}
225
226 \((f \circ g)(x)\) is defined iff
227 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
228
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242
243 \begin{figure*}[ht]
244 \centering
245
246 \begin{tabularx}{\textwidth}{r|Y|Y}
247
248 & \(n\) is even & \(n\) is odd \\ \hline
249
250 \centering \(x^n, n \in \mathbb{Z}^+\) &
251
252 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
253 \begin{axis}[blank, xmin=-3, xmax=3]
254 \addplot[blankplot] {(x^2)};
255 \end{axis}
256 \end{tikzpicture}} &
257
258 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
259 \begin{axis}[blank, xmin=-3, xmax=3]
260 \addplot[blankplot, domain=-3:3] {(x^3)};
261 \end{axis}
262 \end{tikzpicture}} \\ \hline
263
264 \centering \(x^n, n \in \mathbb{Z}^-\) &
265
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267 \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
268 \addplot[blankplot, samples=100] {(x^(-2))};
269 \end{axis}
270 \end{tikzpicture}} &
271
272 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
273 \begin{axis}[blank, xmin=-3, xmax=3]
274 \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
275 \addplot[blankplot, domain=0.1:3] {(x^(-1))};
276 \end{axis}
277 \end{tikzpicture}} \\ \hline
278
279 \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
280
281 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
282 \begin{axis}[blank, xmin=-1, xmax=5]
283 \addplot[blankplot] {(x^(1/2))};
284 \end{axis}
285 \end{tikzpicture}} &
286
287 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
288 \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
289 \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
290 \end{axis}
291 \end{tikzpicture}} \\ \hline
292
293 \end{tabularx}
294 \end{figure*}
295
296 \section{Polynomials}
297
298 \subsection*{Linear equations}
299
300 \subsubsection*{Forms}
301
302 \begin{itemize}
303 \tightlist
304 \item \(y=mx+c\)
305 \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
306 \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
307 \end{itemize}
308
309 \subsubsection*{Line properties}
310
311 Parallel lines: \(m_1 = m_2\)\\
312 Perpendicular lines: \(m_1 \times m_2 = -1\)\\
313 Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
314
315 \subsection*{Quadratics}
316 \setlength{\abovedisplayskip}{1pt}
317 \setlength{\belowdisplayskip}{1pt}
318 \[ x^2 + bx + c = (x+m)(x+n) \]
319 \hfill where \(mn=c, \> m+n=b\)
320
321 \textbf{Difference of squares}
322 \[ a^2 - b^2 = (a-b)(a+b) \]
323 \textbf{Perfect squares}
324 \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
325 \textbf{Completing the square}
326 \begin{align*}
327 x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
328 ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
329 \end{align*}
330 \textbf{Quadratic formula}
331 \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
332 \hfill (Discriminant \(\Delta=b^2-4ac\))
333
334 \subsection*{Cubics}
335
336 \textbf{Difference of cubes}
337 \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
338 \textbf{Sum of cubes}
339 \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
340 \textbf{Perfect cubes}
341 \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
342
343 \[ y=a(bx-h)^3 + c \]
344
345 \begin{itemize}
346 \tightlist
347 \item
348 \(m=0\) at \emph{stationary point of inflection}
349 (i.e.~(\({h \over b}, k)\))
350 \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
351 \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
352 \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
353 \end{itemize}
354
355 \subsection*{Quartic graphs}
356
357 \subsubsection*{Forms of quartic equations}
358
359 \(y=ax^4\)\\
360 \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
361 \(y=ax^4+cd^2 (c \ge 0)\)\\
362 \(y=ax^2(x-b)(x-c)\)\\
363 \(y=a(x-b)^2(x-c)^2\)\\
364 \(y=a(x-b)(x-c)^3\)
365
366 \input{transformations}
367 \input{stuff}
368 \input{circ-functions}
369 \input{calculus}
370
371 \subfile{statistics-ref}
372
373 \end{multicols}
374
375\end{document}