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70\pagestyle{fancy}
71\fancyhead[LO,LE]{Year 12 Methods}
72\fancyhead[CO,CE]{Andrew Lorimer}
73\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
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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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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}
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266
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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
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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}