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69\pagestyle{fancy}
70\fancyhead[LO,LE]{Year 12 Methods}
71\fancyhead[CO,CE]{Andrew Lorimer}
72\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page
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99
100\begin{document}
101
102\title{\vspace{-20mm}Year 12 Methods}
103\author{Andrew Lorimer}
104\date{}
105\maketitle
106
107\begin{multicols}{2}
108
109
110\section{Functions}
111
112\begin{itemize} \tightlist
113 \item vertical line test
114 \item each \(x\) value produces only one \(y\) value
115\end{itemize}
116
117\subsection*{One to one functions}
118
119\begin{itemize} \tightlist
120 \item
121 \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if
122 \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\
123 \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1,
124 \(x^3\) is)
125 \item
126 horizontal line test
127 \item
128 if not one to one, it is many to one
129\end{itemize}
130
131\subsection*{Odd and even functions}
132
133\begin{align*}
134 \text{Even:}&& f(x) &= f(-x) \\
135 \text{Odd:} && -f(x) &= f(-x)
136\end{align*}
137
138Even \(\implies\) symmetrical across \(y\)-axis \\
139\(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\
140For \(x^n\), parity of \(n \equiv\) parity of function
141
142\begin{tabularx}{\columnwidth}{XX}
143 \textbf{Even:} & \textbf{Odd:} \\
144 \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} &
145 \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}
146\end{tabularx}
147
148\subsection*{Inverse functions}
149
150\begin{itemize} \tightlist
151 \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\)
152 \item \(f\) must be one to one
153 \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\)
154 \item Represents reflection across \(y=x\)
155 \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\)
156 \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\
157 \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\)
158 \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\
159\end{itemize}
160
161\subsubsection*{Finding \(f^{-1}\)}
162
163\begin{enumerate} \tightlist
164 \item Let \(y=f(x)\)
165 \item Swap \(x\) and \(y\) (``take inverse''
166 \item Solve for \(y\) \\
167 Sqrt: state \(\pm\) solutions then restrict
168 \item State rule as \(f^{-1}(x)=\dots\)
169 \item For inverse \emph{function}, state in function notation
170\end{enumerate}
171
172\subsection*{Simultaneous equations (linear)}
173
174\begin{itemize} \tightlist
175 \item \textbf{Unique solution} - lines intersect at point
176 \item \textbf{Infinitely many solutions} - lines are equal
177 \item \textbf{No solution} - lines are parallel
178\end{itemize}
179
180\subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions}
181 where all coefficients are known except for one, and \(a, b\) are known
182
183 \begin{enumerate} \tightlist
184 \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\)
185 \item Find determinant of first matrix: \(\Delta = ps-qr\)
186 \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\
187 or let \(\Delta \ne 0\) for one unique solution.
188 \item Solve determinant equation to find variable \\
189 \textbf{For infinite/no solutions:}
190 \item Substitute variable into both original equations
191 \item Rearrange equations so that LHS of each is the same
192 \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\
193 \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns)
194 \end{enumerate}
195
196 \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det}
197
198 \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}
199
200 \begin{itemize} \tightlist
201 \item Use elimination
202 \item Generate two new equations with only two variables
203 \item Rearrange \& solve
204 \item Substitute one variable into another equation to find another variable
205 \end{itemize}
206
207 \subsection*{Piecewise functions}
208
209 \[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\]
210
211 \textbf{Open circle:} point included\\
212 \textbf{Closed circle:} point not included
213
214 \subsection*{Operations on functions}
215
216 For \(f \pm g\) and \(f \times g\):
217 \quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\)
218
219 Addition of linear piecewise graphs: add \(y\)-values at key points
220
221 Product functions:
222
223 \begin{itemize}
224 \tightlist
225 \item
226 product will equal 0 if \(f=0\) or \(g=0\)
227 \item
228 \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\)
229 \end{itemize}
230
231 \subsection*{Composite functions}
232
233 \((f \circ g)(x)\) is defined iff
234 \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
235
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249
250 \begin{figure*}[ht]
251 \centering
252
253 \begin{tabularx}{\textwidth}{r|Y|Y}
254
255 & \(n\) is even & \(n\) is odd \\ \hline
256
257 \centering \(x^n, n \in \mathbb{Z}^+\) &
258
259 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
260 \begin{axis}[blank, xmin=-3, xmax=3]
261 \addplot[blankplot] {(x^2)};
262 \end{axis}
263 \end{tikzpicture}} &
264
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267 \addplot[blankplot, domain=-3:3] {(x^3)};
268 \end{axis}
269 \end{tikzpicture}} \\ \hline
270
271 \centering \(x^n, n \in \mathbb{Z}^-\) &
272
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274 \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
275 \addplot[blankplot, samples=100] {(x^(-2))};
276 \end{axis}
277 \end{tikzpicture}} &
278
279 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
280 \begin{axis}[blank, xmin=-3, xmax=3]
281 \addplot[blankplot, domain=-3:-0.1] {(x^(-1))};
282 \addplot[blankplot, domain=0.1:3] {(x^(-1))};
283 \end{axis}
284 \end{tikzpicture}} \\ \hline
285
286 \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &
287
288 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
289 \begin{axis}[blank, xmin=-1, xmax=5]
290 \addplot[blankplot] {(x^(1/2))};
291 \end{axis}
292 \end{tikzpicture}} &
293
294 \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
295 \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
296 \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
297 \end{axis}
298 \end{tikzpicture}} \\ \hline
299
300 \end{tabularx}
301 \end{figure*}
302
303 \section{Polynomials}
304
305 \subsection*{Linear equations}
306
307 \subsubsection*{Forms}
308
309 \begin{itemize}
310 \tightlist
311 \item \(y=mx+c\)
312 \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph
313 \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts
314 \end{itemize}
315
316 \subsubsection*{Line properties}
317
318 Parallel lines: \(m_1 = m_2\)\\
319 Perpendicular lines: \(m_1 \times m_2 = -1\)\\
320 Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\)
321
322 \subsection*{Quadratics}
323 \setlength{\abovedisplayskip}{1pt}
324 \setlength{\belowdisplayskip}{1pt}
325 \[ x^2 + bx + c = (x+m)(x+n) \]
326 \hfill where \(mn=c, \> m+n=b\)
327
328 \textbf{Difference of squares}
329 \[ a^2 - b^2 = (a-b)(a+b) \]
330 \textbf{Perfect squares}
331 \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \]
332 \textbf{Completing the square}
333 \begin{align*}
334 x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\
335 ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a}
336 \end{align*}
337 \textbf{Quadratic formula}
338 \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \]
339 \hfill (Discriminant \(\Delta=b^2-4ac\))
340
341 \subsection*{Cubics}
342
343 \textbf{Difference of cubes}
344 \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \]
345 \textbf{Sum of cubes}
346 \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]
347 \textbf{Perfect cubes}
348 \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \]
349
350 \[ y=a(bx-h)^3 + c \]
351
352 \begin{itemize}
353 \tightlist
354 \item
355 \(m=0\) at \emph{stationary point of inflection}
356 (i.e.~(\({h \over b}, k)\))
357 \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\)
358 \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\)
359 \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept)
360 \end{itemize}
361
362 \subsection*{Quartic graphs}
363
364 \subsubsection*{Forms of quartic equations}
365
366 \(y=ax^4\)\\
367 \(y=a(x-b)(x-c)(x-d)(x-e)\)\\
368 \(y=ax^4+cd^2 (c \ge 0)\)\\
369 \(y=ax^2(x-b)(x-c)\)\\
370 \(y=a(x-b)^2(x-c)^2\)\\
371 \(y=a(x-b)(x-c)^3\)
372
373 \input{transformations}
374 \input{stuff}
375 \input{circ-functions}
376 \input{calculus}
377
378 \subfile{statistics-ref}
379
380 \end{multicols}
381
382\end{document}