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71\pagestyle{fancy}
72\fancyhead[LO,LE]{Year 12 Methods}
73\fancyhead[CO,CE]{Andrew Lorimer}
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103
104\begin{document}
105
106\title{\vspace{-20mm}Year 12 Methods}
107\author{Andrew Lorimer}
108\date{}
109\maketitle
110
111\begin{multicols}{2}
112
113
114\section{Functions}
115
116\begin{itemize} \tightlist
117 \item vertical line test
118 \item each \(x\) value produces only one \(y\) value
119\end{itemize}
120
121\subsection*{One to one functions}
122
123\begin{itemize} \tightlist
124 \item
125 \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
126 \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
127 \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
128 \(x^3\) is)
129 \item
130 horizontal line test
131 \item
132 if not one to one, it is many to one
133\end{itemize}
134
135\subsection*{Odd and even functions}
136
137\begin{align*}
138 \text{Even:}&& f(x) &= f(-x) \\
139 \text{Odd:} && -f(x) &= f(-x)
140\end{align*}
141
142Even \(\implies\) symmetrical across \(y\)-axis \\
143\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
144For \(x^n\), parity of \(n \equiv\) parity of function
145
146\begin{tabularx}{\columnwidth}{XX}
147 \textbf{Even:} & \textbf{Odd:} \\
148 \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} &
149 \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}
150\end{tabularx}
151
152\subsection*{Inverse functions}
153
154\begin{itemize} \tightlist
155 \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
156 \item \(f\) must be one to one
157 \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
158 \item Represents reflection across \(y=x\)
159 \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
160 \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
161 \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
162 \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
163\end{itemize}
164
165\subsubsection*{Finding \(f^{-1}\)}
166
167\begin{enumerate} \tightlist
168 \item Let \(y=f(x)\)
169 \item Swap \(x\) and \(y\) (``take inverse''
170 \item Solve for \(y\) \\
171 Sqrt: state \(\pm\) solutions then restrict
172 \item State rule as \(f^{-1}(x)=\dots\)
173 \item For inverse \emph{function}, state in function notation
174\end{enumerate}
175
176\subsection*{Simultaneous equations (linear)}
177
178\begin{itemize} \tightlist
179 \item \textbf{Unique solution} - lines intersect at point
180 \item \textbf{Infinitely many solutions} - lines are equal
181 \item \textbf{No solution} - lines are parallel
182\end{itemize}
183
184\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
185 where all coefficients are known except for one, and \(a, b\) are known
186
187 \begin{enumerate} \tightlist
188 \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
189 \item Find \(\det(\text{first matrix}) = ps-qr\)
190 \item Let \(\det = 0\) for \(\{0,\infty\}\) solutions
191 or \(\det \ne 0\) for 1 solution
192 \item Solve to find variable \\ \\
193 \textbf{For infinite/no solutions:}
194 \item Substitute variable into both original equations
195 \item Rearrange so that LHS of each is the same
196 \item \(\begin{aligned}[t]
197 \infty \text{ solns: } & \text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x \\
198 0 \text{ solns: } & \text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x
199 \end{aligned}\)
200 \end{enumerate}
201
202 \begin{cas}
203 Action \(\rightarrow\) Matrix \(\rightarrow\) Calculation \(\rightarrow\) \texttt{det}
204 \end{cas}
205
206 \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
207
208 \begin{itemize} \tightlist
209 \item Use elimination
210 \item Generate two new equations with only two variables
211 \item Rearrange \& solve
212 \item Substitute one variable into another equation to find another variable
213 \end{itemize}
214
215 \subsection*{Piecewise functions}
216
217 \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
218
219 \textbf{Open circle:} point included\\
220 \textbf{Closed circle:} point not included
221
222 \subsection*{Operations on functions}
223
224 For \(f \pm g\) and \(f \times g\):
225 \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
226
227 Addition of linear piecewise graphs: add \(y\)-values at key points
228
229 Product functions:
230
231 \begin{itemize}
232 \tightlist
233 \item
234 product will equal 0 if \(f=0\) or \(g=0\)
235 \item
236 \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
237 \end{itemize}
238
239 \subsection*{Composite functions}
240
241 \((f \circ g)(x)\) is defined iff
242 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
243
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253 }
254 \tikzset{
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256 }
257
258 \begin{figure*}[ht]
259 \centering
260
261 \begin{tabularx}{\textwidth}{r|Y|Y}
262
263 & \(n\) is even & \(n\) is odd \\ \hline
264
265 \centering \(x^n, n \in \mathbb{Z}^+\) &
266
267 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
268 \begin{axis}[blank, xmin=-3, xmax=3]
269 \addplot[blankplot] {(x^2)};
270 \end{axis}
271 \end{tikzpicture}} &
272
273 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
274 \begin{axis}[blank, xmin=-3, xmax=3]
275 \addplot[blankplot, domain=-3:3] {(x^3)};
276 \end{axis}
277 \end{tikzpicture}} \\ \hline
278
279 \centering \(x^n, n \in \mathbb{Z}^-\) &
280
281 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
282 \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
283 \addplot[blankplot, samples=100] {(x^(-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]
289 \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
290 \addplot[blankplot, domain=0.1:3] {(x^(-1))};
291 \end{axis}
292 \end{tikzpicture}} \\ \hline
293
294 \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
295
296 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
297 \begin{axis}[blank, xmin=-1, xmax=5]
298 \addplot[blankplot] {(x^(1/2))};
299 \end{axis}
300 \end{tikzpicture}} &
301
302 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
303 \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
304 \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
305 \end{axis}
306 \end{tikzpicture}} \\ \hline
307
308 \end{tabularx}
309 \end{figure*}
310
311 \section{Polynomials}
312
313 \subsection*{Factor theorem}
314
315 \begin{theorembox}{title=General form \(\beta x + \alpha\)}
316 If \(\beta x + \alpha\) is a factor of \(P(x)\), \\
317 \-\hspace{1em}then \(P(-\dfrac{\alpha}{\beta})=0\).
318 \end{theorembox}
319
320 \begin{theorembox}{title=Simple form \(x-a\)}
321 If \((x-a)\) is a factor of \(P(x)\), remainder \(R=0\). \\
322 \-\hspace{1em}\(\implies P(a)=0\)
323 \end{theorembox}
324
325 \subsection*{Remainder theorem}
326
327 \begin{theorembox}{}
328 When \(P(x)\) is divided by \(\beta x + \alpha\), the remainder is \(-\dfrac{\alpha}{\beta}\).
329 \end{theorembox}
330
331 \subsection*{Rational root theorem}
332 Let \(P(x) = a_n x^n + a_{n-1} x^{n-1} + \dots + a_1 x + a_0\) be a polynomial of degree \(n\) with \(a_i \in \mathbb{Z} \forall a\). Let \(\alpha, \beta \in \mathbb{Z}\) such that their highest common factor is 1 (i.e. relatively prime).
333
334 If \(\beta x + \alpha\) is a factor of \(P(x)\), then \(\beta\) divides \(a_n\) and \(\alpha\) divides \(a_0\) .
335
336 \subsubsection*{Discriminant}
337 \[\begin{cases}
338 b^2-4ac > 0 & \text{two solutions} \\
339 b^2-4ac = 0 & \text{one solution} \\
340 b^2-4ac < 0 & \text{no solutions}
341 \end{cases}\]
342 \begin{warning}
343 Flip inequality sign when multiplying by -1
344 \end{warning}
345
346 \subsection*{Long division}
347
348 \[ \polylongdiv{x^2+2x+4}{x-1} \]
349
350 \begin{cas}
351 Action \(\rightarrow\) Transformation \(\rightarrow\) \texttt{propFrac}
352 \end{cas}
353
354 \subsection*{Linear equations}
355
356 \subsubsection*{Forms}
357
358 \begin{itemize}
359 \tightlist
360 \item \(y=mx+c\)
361 \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
362 \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
363 \end{itemize}
364
365 \subsubsection*{Line properties}
366
367 Parallel lines: \(m_1 = m_2\)\\
368 Perpendicular lines: \(m_1 \times m_2 = -1\)\\
369 Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
370
371 \subsection*{Quadratics}
372
373 \setlength{\abovedisplayskip}{1pt}
374 \setlength{\belowdisplayskip}{1pt}
375
376 \textbf{Linear factorisation}
377 \[ x^2 + bx + c = (x+m)(x+n) \]
378 \hfill where \(mn=c, \> m+n=b\)
379
380 \textbf{Difference of squares}
381 \[ a^2 - b^2 = (a-b)(a+b) \]
382 \textbf{Perfect squares}
383 \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
384 \textbf{Completing the square}
385 \begin{align*}
386 x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
387 ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
388 \end{align*}
389 \textbf{Quadratic formula}
390 \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
391 \hfill (Discriminant \(\Delta=b^2-4ac\))
392
393 \subsection*{Cubics}
394
395 \textbf{Difference of cubes}
396 \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
397 \textbf{Sum of cubes}
398 \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
399 \textbf{Perfect cubes}
400 \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
401
402 \[ y=a(bx-h)^3 + c \]
403
404 \begin{itemize}
405 \tightlist
406 \item
407 \(m=0\) at \emph{stationary point of inflection}
408 (i.e.~(\({h \over b}, k)\))
409 \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
410 \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
411 \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
412 \end{itemize}
413
414 \subsection*{Quartic graphs}
415
416 \subsubsection*{Forms of quartic equations}
417
418 \(y=ax^4\)\\
419 \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
420 \(y=ax^4+cd^2 (c \ge 0)\)\\
421 \(y=ax^2(x-b)(x-c)\)\\
422 \(y=a(x-b)^2(x-c)^2\)\\
423 \(y=a(x-b)(x-c)^3\)
424
425 \input{transformations}
426 \input{stuff}
427 \input{circ-functions}
428 \input{calculus}
429
430 \subfile{statistics-ref}
431
432 \end{multicols}
433
434\end{document}