From: Andrew Lorimer Date: Mon, 12 Aug 2019 04:46:59 +0000 (+1000) Subject: [methods] corrections & layout of collated notes X-Git-Tag: yr12~65 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/d165d17c3c98d1c4cc9507d4aa58de2fd1a1be4c [methods] corrections & layout of collated notes --- diff --git a/methods/calculus.tex b/methods/calculus.tex new file mode 100644 index 0000000..4478faa --- /dev/null +++ b/methods/calculus.tex @@ -0,0 +1,169 @@ +\section{Calculus} + +\subsection*{Average rate of change} + +\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\] + +\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation +\(\rightarrow\) \texttt{diff} + +\subsection*{Average value} + +\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \] + +\subsection*{Instantaneous rate of change} + +\textbf{Secant} - line passing through two points on a curve\\ +\textbf{Chord} - line segment joining two points on a curve + +\subsection*{Limit theorems} + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\tightlist +\item + For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) +\item + \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) +\item + \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) +\item + \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) +\end{enumerate} + +A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). + +\subsection*{First principles derivative} + +\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\] + +Not differentiable at: +\begin{itemize} +\tightlist +\item + discontinuous points +\item + sharp point/cusp +\item + vertical tangents (\(\infty\) gradient) +\end{itemize} + +\subsection*{Tangents \& gradients} + +\textbf{Tangent line} - defined by \(y=mx+c\) where +\(m={dy \over dx}\)\\ +\textbf{Normal line} - \(\perp\) tangent +(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\ +\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\) + +\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation +\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal} + +\subsection*{Strictly increasing/decreasing} + +For \(x_2\) and \(x_1\) where \(x_2 > x_1\): + +\begin{itemize} +\tightlist +\item + \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\) +\item + \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\) +\item + Endpoints are included, even where gradient \(=0\) +\end{itemize} + +\columnbreak + +\subsubsection*{Solving on CAS} + +\colorbox{cas}{\textbf{In main}}: type function. Interactive +\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal +\textbar{} Tan line)\\ +\colorbox{cas}{\textbf{In graph}}: define function. Analysis +\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). +Type \(x\) value to solve for a point. Return to show equation for line. + +\subsection*{Stationary points} + +\begin{align*} + \textbf{Stationary point:} && f^\prime(x) &= 0 \\ + \textbf{Point of inflection:} && f^{\prime\prime} &= 0 +\end{align*} + + \begin{tikzpicture} + \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle] + \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)}; + \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)}; + \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ; + \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ; + \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ; + \end{axis} + \end{tikzpicture}\\ + \begin{tikzpicture} + \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle] + \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)}; + \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)}; + \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ; + \end{axis} + \end{tikzpicture}\\ +\pagebreak +\subsection*{Derivatives} + +\definecolor{shade1}{HTML}{ffffff} +\definecolor{shade2}{HTML}{F0F9E4} +\rowcolors{1}{shade1}{shade2} + \renewcommand{\arraystretch}{1.4} + \begin{tabularx}{\columnwidth}{rX} + \hline + \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\ + \hline + \(\sin x\) & \(\cos x\)\\ + \(\sin ax\) & \(a\cos ax\)\\ + \(\cos x\) & \(-\sin x\)\\ + \(\cos ax\) & \(-a \sin ax\)\\ + \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ + \(e^x\) & \(e^x\)\\ + \(e^{ax}\) & \(ae^{ax}\)\\ + \(ax^{nx}\) & \(an \cdot e^{nx}\)\\ + \(\log_e x\) & \(\dfrac{1}{x}\)\\ + \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ + \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ + \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ + \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ + \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\ + \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ + \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\ + \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\ + \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\ + \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ + \hline + \end{tabularx} + \columnbreak +\subsection*{Antiderivatives} +\rowcolors{1}{shade1}{cas} + \renewcommand{\arraystretch}{1.4} + \begin{tabularx}{\columnwidth}{rX} + \hline + \(f(x)\) & \(\int f(x) \cdot dx\) \\ + \hline + \(k\) (constant) & \(kx + c\)\\ + \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ + \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\ + \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ + \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\ + \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\ + \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ + \(e^k\) & \(e^kx + c\)\\ + \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ + \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\ + \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\ + \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ + \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ + \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ + \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\ + \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ + \hline + \end{tabularx} + diff --git a/methods/circ-functions.tex b/methods/circ-functions.tex index a2d85bf..7580791 100644 --- a/methods/circ-functions.tex +++ b/methods/circ-functions.tex @@ -91,8 +91,8 @@ \begin{tikzpicture} \begin{axis}[yticklabel style={yshift=1.0pt, anchor=north east},x=0.1cm, y=1cm, ymax=2, ymin=-2, xticklabels={}, ytick={-1.5708,1.5708},yticklabels={\(-\frac{\pi}{2}\),\(\frac{\pi}{2}\)}] \addplot[color=orange, smooth] gnuplot [domain=-35:35, unbounded coords=jump,samples=350] {atan(x)} node [pos=0.5, above left] {\(\tan^{-1}x\)}; - \addplot[->, gray, dotted, thick, domain=-35:35] {1.5708}; - \addplot[->, gray, dotted, thick, domain=-35:35] {-1.5708}; + \addplot[gray, dotted, thick, domain=-35:35] {1.5708} node [black, font=\footnotesize, below right, pos=0] {\(y=\frac{\pi}{2}\)}; + \addplot[gray, dotted, thick, domain=-35:35] {-1.5708} node [black, font=\footnotesize, above left, pos=1] {\(y=-\frac{\pi}{2}\)}; \end{axis} \end{tikzpicture} @@ -122,7 +122,7 @@ where: \item Asymptotes at \(x=\frac{(2k+1)\pi}{2n}\) \end{description} -\textbf{Asymptotes should always have equations and arrow pointing up} +\textbf{Asymptotes should always have equations} \subsection*{Solving trig equations} diff --git a/methods/methods-collated.pdf b/methods/methods-collated.pdf index 808c3b5..acdd213 100644 Binary files a/methods/methods-collated.pdf and b/methods/methods-collated.pdf differ diff --git a/methods/methods-collated.tex b/methods/methods-collated.tex index b853626..618f443 100644 --- a/methods/methods-collated.tex +++ b/methods/methods-collated.tex @@ -8,6 +8,7 @@ \usepackage{amssymb} \usepackage{harpoon} \usepackage{tabularx} +\usepackage{tabu} \usepackage{makecell} \usepackage[dvipsnames, table]{xcolor} \usepackage{blindtext} @@ -28,6 +29,7 @@ \pagestyle{fancy} \fancyhead[LO,LE]{Year 12 Methods} \fancyhead[CO,CE]{Andrew Lorimer} +\fancypagestyle{plain}{\fancyhead[LO,LE]{} \fancyhead[CO,CE]{}} % rm title & author for first page \providecommand{\tightlist}{\setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} \setlength{\parindent}{0cm} \usepackage{mathtools} @@ -54,251 +56,241 @@ xlabel={$x$}, % axes labels ylabel={$y$}, }} + \begin{document} -\title{\vspace{-2cm}\hrule\vspace{0.4cm} Year 12 Methods} +\title{\vspace{-20mm}Year 12 Methods} \author{Andrew Lorimer} \date{} \maketitle \begin{multicols}{2} -\section{Functions} - -\begin{itemize} - \tightlist - \item vertical line test - \item each \(x\) value produces only one \(y\) value -\end{itemize} - -\subsection*{One to one functions} + \section{Functions} + + \begin{itemize} + \tightlist + \item vertical line test + \item each \(x\) value produces only one \(y\) value + \end{itemize} + + \subsection*{One to one functions} + + \begin{itemize} \tightlist + \item + \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if + \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ + \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, + \(x^3\) is) + \item + horizontal line test + \item + if not one to one, it is many to one + \end{itemize} + + \subsection*{Odd and even functions} + + \begin{align*} + \text{Even:}&& f(x) &= f(-x) \\ + \text{Odd:} && -f(x) &= f(-x) + \end{align*} + + Even \(\implies\) symmetrical across \(y\)-axis \\ + \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\ + For \(x^n\), parity of \(n \equiv\) parity of function + + \begin{tabularx}{\columnwidth}{XX} + \textbf{Even:} & \textbf{Odd:} \\ + \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} & + \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} + \end{tabularx} + + \subsection*{Inverse functions} + + \begin{itemize} \tightlist + \item Inverse of \(f(x)\) is denoted \(f^{-1}(x)\) + \item \(f\) must be one to one + \item If \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) + \item Represents reflection across \(y=x\) + \item \(\implies f^{-1}(x)=f(x)\) intersections lie on \(y=x\) + \item \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1} \\ + \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) + \item ``Inverse'' \(\ne\) ``inverse \emph{function}'' (functions must pass vertical line test)\\ + \end{itemize} + + \subsubsection*{Finding \(f^{-1}\)} + + \begin{enumerate} \tightlist + \item Let \(y=f(x)\) + \item Swap \(x\) and \(y\) (``take inverse'' + \item Solve for \(y\) \\ + Sqrt: state \(\pm\) solutions then restrict + \item State rule as \(f^{-1}(x)=\dots\) + \item For inverse \emph{function}, state in function notation + \end{enumerate} + + \subsection*{Simultaneous equations (linear)} + + \begin{itemize} \tightlist + \item \textbf{Unique solution} - lines intersect at point + \item \textbf{Infinitely many solutions} - lines are equal + \item \textbf{No solution} - lines are parallel + \end{itemize} + + \subsubsection*{Solving \(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) for \(\{0,1,\infty\}\) solutions} + where all coefficients are known except for one, and \(a, b\) are known + + \begin{enumerate} \tightlist + \item Write as matrices: \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\) + \item Find determinant of first matrix: \(\Delta = ps-qr\) + \item Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\ + or let \(\Delta \ne 0\) for one unique solution. + \item Solve determinant equation to find variable \\ + \textbf{For infinite/no solutions:} + \item Substitute variable into both original equations + \item Rearrange equations so that LHS of each is the same + \item \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) (\(\infty\) solns)\\ + \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 solns) + \end{enumerate} + + \colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} + + \subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)} + + \begin{itemize} \tightlist + \item Use elimination + \item Generate two new equations with only two variables + \item Rearrange \& solve + \item Substitute one variable into another equation to find another variable + \end{itemize} + +\subsection*{Piecewise functions} + +\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\] + +\textbf{Open circle:} point included\\ +\textbf{Closed circle:} point not included + +\subsection*{Operations on functions} + +For \(f \pm g\) and \(f \times g\): +\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\) + +Addition of linear piecewise graphs: add \(y\)-values at key points + +Product functions: \begin{itemize} \tightlist \item - \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if - \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ - \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, - \(x^3\) is) + product will equal 0 if \(f=0\) or \(g=0\) \item - horizontal line test -\item - if not one to one, it is many to one + \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\) \end{itemize} -\subsection*{Finding inverse functions \(f^{-1}\)} - -\begin{itemize} -\tightlist -\item - if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) -\item - reflection across \(y-x\) -\item - \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) -\item - inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass - vertical line test)\\ - \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one -\item - \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\) -\end{itemize} - -\subsubsection*{Requirements for showing working for \(f^{-1}\)} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - start with \emph{``let \(y=f(x)\)''} -\item - must state \emph{``take inverse''} for line where \(y\) and \(x\) are - swapped -\item - do all working in terms of \(y=\dots\) -\item - for sqrt, state \(\pm\) solutions then show restricted -\item - for inverse \emph{function}, state in function notation -\end{enumerate} -\subsubsection*{Solving -\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) -for \(\{0,1,\infty\}\) -solutions} - -where all coefficients are known except for one, and \(a, b\) are known - -\begin{enumerate} -\tightlist -\item - Write as matrices: - \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\) -\item - Find determinant of first matrix: \(\Delta = ps-qr\) -\item - Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\ - or let \(\Delta \ne 0\) for one unique solution. -\item - Solve determinant equation to find variable \\ - \textbf{For infinite/no solutions:} -\item - Substitute variable into both original equations -\item - Rearrange equations so that LHS of each is the same -\item - \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) - (\(\infty\) solns)\\ - \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 - solns) -\end{enumerate} - -\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} - -\subsubsection*{Solving \(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)} - -\begin{itemize} -\tightlist -\item - Use elimination -\item - Generate two new equations with only two variables -\item - Rearrange \& solve -\item - Substitute one variable into another equation to find another variable -\end{itemize} -\subsection*{Odd and even functions} - -Even when \(f(x) = -f(x)\)\\ -Odd when \(-f(x) = f(-x)\) - -Function is even if it is symmetrical across \(y\)-axis -\hspace{5em}\(\implies f(x)=f(-x)\)\\ -Function \(x^{\pm {p \over q}}\) is odd if \(q\) is odd\\ - -\begin{tabularx}{\columnwidth}{XX} - \textbf{Even:} & \textbf{Odd:} \\ - \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} & - \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} -\end{tabularx} -\pagebreak - \pgfplotsset{every axis/.append style={ - xlabel=, % put the x axis in the middle - ylabel=, % put the y axis in the middle - }} - \begin{table*}[ht] - \centering - \begin{tabularx}{\textwidth}{r|X|X} - & \(n\) is even & \(n\) is odd \\ \hline - \(x^n, n \in \mathbb{Z}^+\) & - \makecell{\\\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}} & - \makecell{\\\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}} \\ - \(x^n, n \in \mathbb{Z}^-\) & - \makecell{\\\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}} & - \makecell{\\\begin{tikzpicture}\begin{axis}[yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, scale=0.4, samples=100, smooth] \addplot[orange, mark=none] {(x^(-1))}; \end{axis}\end{tikzpicture}} \\ - \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) & - \makecell{\\\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}} & - \makecell{\\\begin{tikzpicture} +\subsection*{Composite functions} + +\((f \circ g)(x)\) is defined iff +\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) + + + \pgfplotsset{every axis/.append style={ ticks=none, xlabel=, ylabel=, }} % remove axis labels & ticks + \begin{table*}[ht] + \centering + \begin{tabu} to \textwidth {@{} X[0.3,r] *2{|X[c,m]}@{}} + & \(n\) is even & \(n\) is odd \\ \tabucline{1pt} + \(x^n, n \in \mathbb{Z}^+\) & + \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} & + \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} \\ + \(x^n, n \in \mathbb{Z}^-\) & + \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} & + \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} \\ + \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) & + \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} & + \begin{tikzpicture} \begin{axis}[enlargelimits=false, yticklabels={,,}, xticklabels={,,}, xmin=-3, xmax=3, ymin=-3, ymax=3, smooth, scale=0.4] -\addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) }; -\end{axis} - \end{tikzpicture}} - \end{tabularx} - \end{table*} - \pgfplotsset{every axis/.append style={ - xlabel=\(x\), % put the x axis in the middle - ylabel=\(y\), % put the y axis in the middle - }} - -\section{Polynomials} - -\subsection*{Quadratics} - -\[ x^2 + bx + c = (x+m)(x+n) \] -\hfill where \(mn=c, \> m+n=b\) - -\begin{align*} - \hline - \textbf{Difference} && a^2 - b^2 &= (a-b)(a+b) \\[2ex] - \textbf{Perfect sq.} && a^2 \pm 2ab + b^2 &= (a \pm b^2) \\[2ex] - \textbf{Completing} && x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\ - && ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a} \\[2ex] - \textbf{Quadratic} && x &= \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \\ - && & \text{where} \Delta=b^2-4ac \\ - \hline -\end{align*} - -\subsection*{Cubics} - -\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\ -\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\ -\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\) - -\[ y=a(bx-h)^3 + c \] - -\begin{itemize} -\tightlist -\item - \(m=0\) at \emph{stationary point of inflection} - (i.e.~(\({h \over b}, k)\)) -\item - in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\) -\item - in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\) -\item - in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at - \(c\) -\end{itemize} - -\subsection*{Linear and quadratic -graphs} - -\subsubsection*{Forms of linear -equations} - -\begin{itemize} -\tightlist - \item \(y=mx+c\) - \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph - \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts -\end{itemize} - -\subsection*{Line properties} - -Parallel lines: \(m_1 = m_2\)\\ -Perpendicular lines: \(m_1 \times m_2 = -1\)\\ -Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) - -\subsection*{Quartic graphs} - -\subsubsection*{Forms of quartic -equations} - -\(y=ax^4\)\\ -\(y=a(x-b)(x-c)(x-d)(x-e)\)\\ -\(y=ax^4+cd^2 (c \ge 0)\)\\ -\(y=ax^2(x-b)(x-c)\)\\ -\(y=a(x-b)^2(x-c)^2\)\\ -\(y=a(x-b)(x-c)^3\) - -\subsection*{Simultaneous equations -(linear)} - -\begin{itemize} -\tightlist -\item - \textbf{Unique solution} - lines intersect at point -\item - \textbf{Infinitely many solutions} - lines are equal -\item - \textbf{No solution} - lines are parallel -\end{itemize} - - -\input{temp/transformations} -\input{temp/stuff} -\input{circ-functions} -\input{temp/calculus} - -\end{multicols} -\end{document} + \addplot [orange,domain=-2:2,samples=1000,no markers] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) }; + \end{axis} + \end{tikzpicture} + \end{tabu} + \hrule + \end{table*} + \pgfplotsset{every axis/.append style={ xlabel=\(x\), ylabel=\(y\) }} % put axis labels back + + \section{Polynomials} + + \subsection*{Linear equations} + + \subsubsection*{Forms} + + \begin{itemize} + \tightlist + \item \(y=mx+c\) + \item \(\frac{x}{a} + \frac{y}{b}=1\) where \((x_1, y_1)\) lies on the graph + \item \(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and \(y\)-intercepts + \end{itemize} + + \subsubsection*{Line properties} + + Parallel lines: \(m_1 = m_2\)\\ + Perpendicular lines: \(m_1 \times m_2 = -1\)\\ + Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) + + \subsection*{Quadratics} + \setlength{\abovedisplayskip}{1pt} + \setlength{\belowdisplayskip}{1pt} + \[ x^2 + bx + c = (x+m)(x+n) \] + \hfill where \(mn=c, \> m+n=b\) + + \textbf{Difference of squares} + \[ a^2 - b^2 = (a-b)(a+b) \] + \textbf{Perfect squares} + \[ a^2 \pm 2ab + b^2 = (a \pm b^2) \] + \textbf{Completing the square} + \begin{align*} + x^2+bx+c &= (x+\frac{b}{2})^2+c-\frac{b^2}{4} \\ + ax^2+bx+c &= a(x-\frac{b}{2a})^2+c-\frac{b^2}{4a} + \end{align*} + \textbf{Quadratic formula} + \[ x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a} \] + \hfill (Discriminant \(\Delta=b^2-4ac\)) + + \subsection*{Cubics} + + \textbf{Difference of cubes} + \[ a^3 - b^3 = (a-b)(a^2 + ab + b^2) \] + \textbf{Sum of cubes} + \[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \] + \textbf{Perfect cubes} + \[ a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3 \] + + \[ y=a(bx-h)^3 + c \] + + \begin{itemize} + \tightlist + \item + \(m=0\) at \emph{stationary point of inflection} + (i.e.~(\({h \over b}, k)\)) + \item \(y=(x-a)^2(x-b)\) --- max at \(x=a\), min at \(x=b\) + \item \(y=a(x-b)(x-c)(x-d)\) --- roots at \(b, c, d\) + \item \(y=a(x-b)^2(x-c)\) --- roots at \(b\) (instantaneous), \(c\) (intercept) + \end{itemize} + + \subsection*{Quartic graphs} + + \subsubsection*{Forms of quartic equations} + + \(y=ax^4\)\\ + \(y=a(x-b)(x-c)(x-d)(x-e)\)\\ + \(y=ax^4+cd^2 (c \ge 0)\)\\ + \(y=ax^2(x-b)(x-c)\)\\ + \(y=a(x-b)^2(x-c)^2\)\\ + \(y=a(x-b)(x-c)^3\) + + \input{transformations} + \input{stuff} + \input{circ-functions} + \input{calculus} + + \end{multicols} + \end{document} diff --git a/methods/stuff.tex b/methods/stuff.tex new file mode 100644 index 0000000..071a828 --- /dev/null +++ b/methods/stuff.tex @@ -0,0 +1,118 @@ +\section{Exponentials \& Logarithms} + +\subsubsection*{Logarithmic identities} + +\begin{align*} + \log_b (xy) &= \log_b x + \log_b y \\ + \log_b x^n &= n \log_b x \\ + \log_b y^{x^n} &= x^n \log_b y \\ + \log_a(\frac{m}{n}) &= \log_am - \log_a \\ + \log_a(m^{-1}) & = -\log_am \\ + \log_b c &= \frac{\log_a c}{\log_a b} +\end{align*} + +\subsubsection*{Index identities} + +\begin{align*} + b^{m+n} &= b^m \cdot b^n \\ + (b^m)^n &= b^{m \cdot n} \\ + (b \cdot c)^n &= b^n \cdot c^n \\ + {b^m \div a^n} &= {b^{m-n}} +\end{align*} + +\subsection*{Inverse functions} + +For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is: + +\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\] + +\subsection*{Euler's number \(e\)} + +\[e= \lim_{n \rightarrow \infty} \left(1 + \dfrac{1}{n}\right)^n\] + +\subsection*{Modelling} + +\[A = A_0 e^{kt}\] + +\begin{itemize} +\tightlist +\item + \(A_0\) is initial value +\item + \(t\) is time taken +\item + \(k\) is a constant +\item + For continuous growth, \(k > 0\) +\item + For continuous decay, \(k < 0\) +\end{itemize} + +\subsection*{Graphing exponential functions} + +\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\] + +\begin{itemize} +\tightlist +\item + \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as + \(x \rightarrow \infty\) +\item + \textbf{horizontal asymptote} at \(y=c\) +\item + \textbf{domain} is \(\mathbb{R}\) +\item + \textbf{range} is \((c, \infty)\) +\item + dilation of factor \(|A|\) from \(x\)-axis +\item + dilation of factor \(1 \over k\) from \(y\)-axis +\end{itemize} + +\begin{tikzpicture} + \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none] + \addplot[red, thick, smooth, samples=100] plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)}; + \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)}; + \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)}; + \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ; + \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)}; + \end{axis} +\end{tikzpicture} + +\subsection*{Graphing logarithmic functions} + +\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\)) + +\[f(x)=A \log_a k(x-b) + c\] + +where + +\begin{itemize} +\tightlist +\item + \textbf{domain} is \((b, \infty)\) +\item + \textbf{range} is \(\mathbb{R}\) +\item + \textbf{vertical asymptote} at \(x=b\) +\item + \(y\)-intercept exists if \(b<0\) +\item + dilation of factor \(|A|\) from \(x\)-axis +\item + dilation of factor \(1 \over k\) from \(y\)-axis +\end{itemize} +\begin{tikzpicture} + \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none] + \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)}; + \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)}; + \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)}; + \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)}; + \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ; + \end{axis} +\end{tikzpicture} + +\subsection*{Finding equations} + +\colorbox{cas}{On CAS:} +\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png} diff --git a/methods/temp/calculus.tex b/methods/temp/calculus.tex deleted file mode 100644 index 541a8c0..0000000 --- a/methods/temp/calculus.tex +++ /dev/null @@ -1,169 +0,0 @@ -\section{Calculus} - -\subsection*{Average rate of change} - -\[m \operatorname{of} x \in [a,b] = \dfrac{f(b)-f(a)}{b - a} = \frac{dy}{dx}\] - -\colorbox{cas}{On CAS:} Action \(\rightarrow\) Calculation -\(\rightarrow\) \texttt{diff} - -\subsection*{Average value} - -\[ f_{\text{avg}} = \dfrac{1}{b-a} \int^b_a f(x) \> dx \] - -\subsection*{Instantaneous rate of change} - -\textbf{Secant} - line passing through two points on a curve\\ -\textbf{Chord} - line segment joining two points on a curve - -\subsection*{Limit theorems} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - For constant function \(f(x)=k\), \(\lim_{x \rightarrow a} f(x) = k\) -\item - \(\lim_{x \rightarrow a} (f(x) \pm g(x)) = F \pm G\) -\item - \(\lim_{x \rightarrow a} (f(x) \times g(x)) = F \times G\) -\item - \({\lim_{x \rightarrow a} {f(x) \over g(x)}} = {F \over G}, G \ne 0\) -\end{enumerate} - -A function is continuous if \(L^-=L^+=f(x)\) for all values of \(x\). - -\subsection*{First principles derivative} - -\[f^\prime(x)=\lim_{h \rightarrow 0}{{f(x+h)-f(x)} \over h}\] - -Not differentiable at: -\begin{itemize} -\tightlist -\item - discontinuous points -\item - sharp point/cusp -\item - vertical tangents (\(\infty\) gradient) -\end{itemize} - -\subsection*{Tangents \& gradients} - -\textbf{Tangent line} - defined by \(y=mx+c\) where -\(m={dy \over dx}\)\\ -\textbf{Normal line} - \(\perp\) tangent -(\(m_{{tan}} \cdot m_{\operatorname{norm}} = -1\))\\ -\textbf{Secant} \(={{f(x+h)-f(x)} \over h}\) - -\colorbox{cas}{On CAS:} \\ Action \(\rightarrow\) Calculation -\(\rightarrow\) Line \(\rightarrow\) \texttt{tanLine} or \texttt{normal} - -\subsection*{Strictly increasing/decreasing} - -For \(x_2\) and \(x_1\) where \(x_2 > x_1\): - -\begin{itemize} -\tightlist -\item - \textbf{strictly increasing}\\ where \(f(x_2) > f(x_1)\) or \(f^\prime(x)>0\) -\item - \textbf{strictly decreasing}\\ where \(f(x_2) < f(x_1)\) or \(f^\prime(x)<0\) -\item - Endpoints are included, even where gradient \(=0\) -\end{itemize} - -\columnbreak - -\subsubsection*{Solving on CAS} - -\colorbox{cas}{\textbf{In main}}: type function. Interactive -\(\rightarrow\) Calculation \(\rightarrow\) Line \(\rightarrow\) (Normal -\textbar{} Tan line)\\ -\colorbox{cas}{\textbf{In graph}}: define function. Analysis -\(\rightarrow\) Sketch \(\rightarrow\) (Normal \textbar{} Tan line). -Type \(x\) value to solve for a point. Return to show equation for line. - -\subsection*{Stationary points} - -\emph{Stationary point} - i.e. -\(f^\prime(x)=0\)\\ -\emph{Point of inflection} - max \(|\)gradient\(|\) (i.e. -\(f^{\prime\prime} = 0\)) - - \begin{tikzpicture} - \begin{axis}[xmin=-21, xmax=21, ymax=1400, ymin=-1000, ticks=none, axis lines=middle] - \addplot[color=red, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {x^3-3*x^2-144*x+432} node [black, pos=1, right] {\(f(x)\)}; - \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=-15:15,unbounded coords=jump,samples=500] {3*x^2-6*x-144} node [black, pos=1, right] {\(f^\prime(x)\)}; - \addplot[mark=*, blue] coordinates {(1,286)} node[above right, align=left, font=\footnotesize]{inflection \\ (falling)} ; - \addplot[mark=*, orange] coordinates {(-6,972)} node[above left, align=right, font=\footnotesize]{stationary \\ (local max)} ; - \addplot[mark=*, orange] coordinates {(8,-400)} node[below, align=left, font=\footnotesize]{stationary \\ (local min)} ; - \end{axis} - \end{tikzpicture}\\ - \begin{tikzpicture} - \begin{axis}[enlargelimits=true, xmax=3.5, ticks=none, axis lines=middle] - \addplot[color=blue, smooth, thick] gnuplot [domain=0.74:3,unbounded coords=jump,samples=500] {(x-2)^3+2} node [black, pos=0.9, left] {\(f(x)\)}; - \addplot[color=darkgray, dashed, smooth, thick] gnuplot [domain=1:3,unbounded coords=jump,samples=500] {3*(x-2)^2} node [black, pos=0.9, right] {\(f^\prime(x)\)}; - \addplot[mark=*, purple] coordinates {(2,2)} node[below right, align=left, font=\footnotesize]{stationary \\ inflection} ; - \end{axis} - \end{tikzpicture}\\ -\pagebreak -\subsection*{Derivatives} - -\definecolor{shade1}{HTML}{ffffff} -\definecolor{shade2}{HTML}{F0F9E4} -\rowcolors{1}{shade1}{shade2} - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \hspace{6em}\(f(x)\) & \(f^\prime(x)\)\\ - \hline - \(\sin x\) & \(\cos x\)\\ - \(\sin ax\) & \(a\cos ax\)\\ - \(\cos x\) & \(-\sin x\)\\ - \(\cos ax\) & \(-a \sin ax\)\\ - \(\tan f(x)\) & \(f^2(x) \sec^2f(x)\)\\ - \(e^x\) & \(e^x\)\\ - \(e^{ax}\) & \(ae^{ax}\)\\ - \(ax^{nx}\) & \(an \cdot e^{nx}\)\\ - \(\log_e x\) & \(\dfrac{1}{x}\)\\ - \(\log_e {ax}\) & \(\dfrac{1}{x}\)\\ - \(\log_e f(x)\) & \(\dfrac{f^\prime (x)}{f(x)}\)\\ - \(\sin(f(x))\) & \(f^\prime(x) \cdot \cos(f(x))\)\\ - \(\sin^{-1} x\) & \(\dfrac{1}{\sqrt{1-x^2}}\)\\ - \(\cos^{-1} x\) & \(\dfrac{-1}{\sqrt{1-x^2}}\)\\ - \(\tan^{-1} x\) & \(\dfrac{1}{1 + x^2}\)\\ - \(\frac{d}{dy}f(y)\) & \(\dfrac{1}{\frac{dx}{dy}}\) \hfill(reciprocal)\\ - \(uv\) & \(u \frac{dv}{dx}+v\frac{du}{dx}\) \hfill(product rule)\\ - \(\dfrac{u}{v}\) & \(\dfrac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}\) \hfill(quotient rule)\\ - \(f(g(x))\) & \(f^\prime(g(x))\cdot g^\prime(x)\)\\ - \hline - \end{tabularx} - \columnbreak -\subsection*{Antiderivatives} -\rowcolors{1}{shade1}{cas} - \renewcommand{\arraystretch}{1.4} - \begin{tabularx}{\columnwidth}{rX} - \hline - \(f(x)\) & \(\int f(x) \cdot dx\) \\ - \hline - \(k\) (constant) & \(kx + c\)\\ - \(x^n\) & \(\dfrac{1}{n+1} x^{n+1}\) \\ - \(a x^{-n}\) &\(a \cdot \log_e |x| + c\)\\ - \(\dfrac{1}{ax+b}\) &\(\dfrac{1}{a} \log_e (ax+b) + c\)\\ - \((ax+b)^n\) & \(\dfrac{1}{a(n+1)}(ax+b)^{n-1} + c\>|\>n\ne 1\)\\ - \((ax+b)^{-1}\) & \(\dfrac{1}{a}\log_e |ax+b|+c\)\\ - \(e^{kx}\) & \(\dfrac{1}{k} e^{kx} + c\)\\ - \(e^k\) & \(e^kx + c\)\\ - \(\sin kx\) & \(\dfrac{-1}{k} \cos (kx) + c\)\\ - \(\cos kx\) & \(\dfrac{1}{k} \sin (kx) + c\)\\ - \(\sec^2 kx\) & \(\dfrac{1}{k} \tan(kx) + c\)\\ - \(\dfrac{1}{\sqrt{a^2-x^2}}\) & \(\sin^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\dfrac{-1}{\sqrt{a^2-x^2}}\) & \(\cos^{-1} \dfrac{x}{a} + c \>\vert\> a>0\)\\ - \(\frac{a}{a^2-x^2}\) & \(\tan^{-1} \frac{x}{a} + c\)\\ - \(\frac{f^\prime (x)}{f(x)}\) & \(\log_e f(x) + c\)\\ - \(\int f(u) \cdot \frac{du}{dx} \cdot dx\) & \(\int f(u) \cdot du\) \hfill(substitution)\\ - \(f(x) \cdot g(x)\) & \(\int [f^\prime(x) \cdot g(x)] dx + \int [g^\prime(x) f(x)] dx\)\\ - \hline - \end{tabularx} - diff --git a/methods/temp/inverse.tex b/methods/temp/inverse.tex deleted file mode 100644 index f02a4d1..0000000 --- a/methods/temp/inverse.tex +++ /dev/null @@ -1,72 +0,0 @@ -\setstretch{1.3} -\pagenumbering{gobble} - -\hypertarget{inverse-functions}{% -\section{Inverse functions}\label{inverse-functions}} - -\hypertarget{functions}{% -\subsection{Functions}\label{functions}} - -\begin{itemize} -\tightlist -\item - vertical line test -\item - each \(x\) value produces only one \(y\) value -\end{itemize} - -\hypertarget{one-to-one-functions}{% -\subsection{One to one functions}\label{one-to-one-functions}} - -\begin{itemize} -\tightlist -\item - \(f(x)\) is \emph{one to one} if \(f(a) \ne f(b)\) if - \(a, b \in \operatorname{dom}(f)\) and \(a \ne b\)\\ - \(\implies\) unique \(y\) for each \(x\) (\(\sin x\) is not 1:1, - \(x^3\) is) -\item - horizontal line test -\item - if not one to one, it is many to one -\end{itemize} - -\hypertarget{deriving-f-1}{% -\subsection{\texorpdfstring{Deriving -\(f^{-1}\)}{Deriving f\^{}\{-1\}}}\label{deriving-f-1}} - -\begin{itemize} -\tightlist -\item - if \(f(g(x)) = x\), then \(g\) is the inverse of \(f\) -\item - reflection across \(y-x\) -\item - \(\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}\) -\item - inverse \(\ne\) inverse \emph{function} (i.e.~inverse must pass - vertical line test)\\ - \(\implies f^{-1}(x)\) exists \(\iff f(x)\) is one to one -\item - \(f^{-1}(x)=f(x)\) intersections may lie on line \(y=x\) -\end{itemize} - -\hypertarget{requirements-for-showing-working-for-f-1}{% -\subsubsection{\texorpdfstring{Requirements for showing working for -\(f^{-1}\)}{Requirements for showing working for f\^{}\{-1\}}}\label{requirements-for-showing-working-for-f-1}} - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - start with \emph{``let \(y=f(x)\)''} -\item - must state \emph{``take inverse''} for line where \(y\) and \(x\) are - swapped -\item - do all working in terms of \(y=\dots\) -\item - for square root, state \(\pm\) solutions then show restricted -\item - for inverse \emph{function}, state in function notation -\end{enumerate} diff --git a/methods/temp/polynomials.tex b/methods/temp/polynomials.tex deleted file mode 100644 index a24f1c0..0000000 --- a/methods/temp/polynomials.tex +++ /dev/null @@ -1,211 +0,0 @@ -% Options for packages loaded elsewhere -\PassOptionsToPackage{unicode}{hyperref} -\PassOptionsToPackage{hyphens}{url} -% -\documentclass[ -]{article} -\usepackage{lmodern} -\usepackage{amssymb,amsmath} -\usepackage{ifxetex,ifluatex} -\ifnum 0\ifxetex 1\fi\ifluatex 1\fi=0 % if pdftex - \usepackage[T1]{fontenc} - \usepackage[utf8]{inputenc} - \usepackage{textcomp} % provide euro and other symbols -\else % if luatex or xetex - \usepackage{unicode-math} - \defaultfontfeatures{Scale=MatchLowercase} - \defaultfontfeatures[\rmfamily]{Ligatures=TeX,Scale=1} -\fi -% Use upquote if available, for straight quotes in verbatim environments -\IfFileExists{upquote.sty}{\usepackage{upquote}}{} -\IfFileExists{microtype.sty}{% use microtype if available - \usepackage[]{microtype} - \UseMicrotypeSet[protrusion]{basicmath} % disable protrusion for tt fonts -}{} -\makeatletter -\@ifundefined{KOMAClassName}{% if non-KOMA class - \IfFileExists{parskip.sty}{% - \usepackage{parskip} - }{% else - \setlength{\parindent}{0pt} - \setlength{\parskip}{6pt plus 2pt minus 1pt}} -}{% if KOMA class - \KOMAoptions{parskip=half}} -\makeatother -\usepackage{xcolor} -\IfFileExists{xurl.sty}{\usepackage{xurl}}{} % add URL line breaks if available -\IfFileExists{bookmark.sty}{\usepackage{bookmark}}{\usepackage{hyperref}} -\hypersetup{ - pdfauthor={Andrew Lorimer}, - hidelinks, - pdfcreator={LaTeX via pandoc}} -\urlstyle{same} % disable monospaced font for URLs -\usepackage[a4paper, margin=2cm]{geometry} -\setlength{\emergencystretch}{3em} % prevent overfull lines -\providecommand{\tightlist}{% - \setlength{\itemsep}{0pt}\setlength{\parskip}{0pt}} -\setcounter{secnumdepth}{-\maxdimen} % remove section numbering -\usepackage{setspace} -\usepackage{fancyhdr} -\pagestyle{fancy} -\fancyhead[LO,LE]{Year 12 Methods} -\fancyhead[CO,CE]{Andrew Lorimer} -\usepackage{graphicx} -\usepackage{tabularx} -\usepackage[dvipsnames]{xcolor} - -\author{Andrew Lorimer} -\date{} - -\begin{document} - -\hypertarget{polynomials}{% -\section{Polynomials}\label{polynomials}} - -\hypertarget{quadratics}{% -\subsection{Quadratics}\label{quadratics}} - -\newcolumntype{R}{>{\raggedleft\arraybackslash}X} -\begin{tabularx}{\columnwidth}{Rl} - General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\ - \hline - Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\ - \hline - Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\ - \hline - Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\ - \hline - Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\ -\end{tabularx} - -\hypertarget{cubics}{% -\subsection{Cubics}\label{cubics}} - -\textbf{Difference of cubes:} \(a^3 - b^3 = (a-b)(a^2 + ab + b^2)\)\\ -\textbf{Sum of cubes:} \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)\\ -\textbf{Perfect cubes:} \(a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3\) - -\[y=a(bx-h)^3 + c\] - -\begin{itemize} -\tightlist -\item - \(m=0\) at \emph{stationary point of inflection} - (i.e.~(\({h \over b}, k)\)) -\item - in form \(y=(x-a)^2(x-b)\), local max at \(x=a\), local min at \(x=b\) -\item - in form \(y=a(x-b)(x-c)(x-d)\): \(x\)-intercepts at \(b, c, d\) -\item - in form \(y=a(x-b)^2(x-c)\), touches \(x\)-axis at \(b\), intercept at - \(c\) -\end{itemize} - -\hypertarget{linear-and-quadratic-graphs}{% -\subsection{Linear and quadratic -graphs}\label{linear-and-quadratic-graphs}} - -\hypertarget{forms-of-linear-equations}{% -\subsubsection{Forms of linear -equations}\label{forms-of-linear-equations}} - -\(y=mx+c\) where \(m\) is gradient and \(c\) is \(y\)-intercept\\ -\({x \over a} + {y \over b}=1\) where \(m\) is gradient and -\((x_1, y_1)\) lies on the graph\\ -\(y-y_1 = m(x-x_1)\) where \((a,0)\) and \((0,b)\) are \(x\)- and -\(y\)-intercepts - -\hypertarget{line-properties}{% -\subsection{Line properties}\label{line-properties}} - -Parallel lines: \(m_1 = m_2\)\\ -Perpendicular lines: \(m_1 \times m_2 = -1\)\\ -Distance: \(|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}\) - -\hypertarget{quartic-graphs}{% -\subsection{Quartic graphs}\label{quartic-graphs}} - -\hypertarget{forms-of-quadratic-equations}{% -\subsubsection{Forms of quadratic -equations}\label{forms-of-quadratic-equations}} - -\(y=ax^4\)\\ -\(y=a(x-b)(x-c)(x-d)(x-e)\)\\ -\(y=ax^4+cd^2 (c \ge 0)\)\\ -\(y=ax^2(x-b)(x-c)\)\\ -\(y=a(x-b)^2(x-c)^2\)\\ -\(y=a(x-b)(x-c)^3\) - -\hypertarget{simultaneous-equations-linear}{% -\subsection{Simultaneous equations -(linear)}\label{simultaneous-equations-linear}} - -\begin{itemize} -\tightlist -\item - \textbf{Unique solution} - lines intersect at point -\item - \textbf{Infinitely many solutions} - lines are equal -\item - \textbf{No solution} - lines are parallel -\end{itemize} - -\hypertarget{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}{% -\subsubsection{\texorpdfstring{Solving -\(\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>\) -for \(\{0,1,\infty\}\) -solutions}{Solving \textbackslash protect\textbackslash begin\{cases\}px + qy = a \textbackslash\textbackslash{} rx + sy = b\textbackslash protect\textbackslash end\{cases\} \textbackslash\textgreater{} for \textbackslash\{0,1,\textbackslash infty\textbackslash\} solutions}}\label{solving-protectbegincasespx-qy-a-rx-sy-bprotectendcases-for-01infty-solutions}} - -where all coefficients are known except for one, and \(a, b\) are known - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\tightlist -\item - Write as matrices: - \(\begin{bmatrix}p & q \\ r & s \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} a \\ b \end{bmatrix}\) -\item - Find determinant of first matrix: \(\Delta = ps-qr\) -\item - Let \(\Delta = 0\) for number of solutions \(\ne 1\)\\ - or let \(\Delta \ne 0\) for one unique solution. -\item - Solve determinant equation to find variable - - \begin{itemize} - \tightlist - \item - \emph{--- for infinite/no solutions: ---} - \end{itemize} -\item - Substitute variable into both original equations -\item - Rearrange equations so that LHS of each is the same -\item - \(\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x\) - (\(\infty\) solns)\\ - \(\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x\) (0 - solns) -\end{enumerate} - -\colorbox{cas}{On CAS:} Matrix \(\rightarrow\) \texttt{det} - -\hypertarget{solving-protectbegincasesa_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_3protectendcases}{% -\subsubsection{\texorpdfstring{Solving -\(\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}\)}{Solving \textbackslash protect\textbackslash begin\{cases\}a\_1 x + b\_1 y + c\_1 z = d\_1 \textbackslash\textbackslash{} a\_2 x + b\_2 y + c\_2 z = d\_2 \textbackslash\textbackslash{} a\_3 x + b\_3 y + c\_3 z = d\_3\textbackslash protect\textbackslash end\{cases\}}}\label{solving-protectbegincasesa_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_3protectendcases}} - -\begin{itemize} -\tightlist -\item - Use elimination -\item - Generate two new equations with only two variables -\item - Rearrange \& solve -\item - Substitute one variable into another equation to find another variable -\item - etc. -\end{itemize} - -\end{document} diff --git a/methods/temp/stuff.tex b/methods/temp/stuff.tex deleted file mode 100644 index 2964bea..0000000 --- a/methods/temp/stuff.tex +++ /dev/null @@ -1,118 +0,0 @@ -\section{Exponentials \& Logarithms} - -\subsubsection*{Logarithmic identities} - -\begin{align*} - \log_b (xy) &= \log_b x + \log_b y \\ - \log_b x^n &= n \log_b x \\ - \log_b y^{x^n} &= x^n \log_b y \\ - \log_a(\frac{m}{n}) &= \log_am - \log_a \\ - \log_a(m^{-1}) & = -\log_am \\ - \log_b c &= \frac{\log_a c}{\log_a b} -\end{align*} - -\subsubsection*{Index identities} - -\begin{align*} - b^{m+n} &= b^m \cdot b^n \\ - (b^m)^n &= b^{m \cdot n} \\ - (b \cdot c)^n &= b^n \cdot c^n \\ - {b^m \div a^n} &= {b^{m-n}} -\end{align*} - -\subsection*{Inverse functions} - -For \(f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x\), inverse is: - -\[f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax\] - -\subsection*{Euler's number \(e\)} - -\[e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n\] - -\subsection*{Modelling} - -\[A = A_0 e^{kt}\] - -\begin{itemize} -\tightlist -\item - \(A_0\) is initial value -\item - \(t\) is time taken -\item - \(k\) is a constant -\item - For continuous growth, \(k > 0\) -\item - For continuous decay, \(k < 0\) -\end{itemize} - -\subsection*{Graphing exponential functions} - -\[f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1\] - -\begin{itemize} -\tightlist -\item - \textbf{\(y\)-intercept} at \((0, A \cdot a^{-kb}+c)\) as - \(x \rightarrow \infty\) -\item - \textbf{horizontal asymptote} at \(y=c\) -\item - \textbf{domain} is \(\mathbb{R}\) -\item - \textbf{range} is \((c, \infty)\) -\item - dilation of factor \(|A|\) from \(x\)-axis -\item - dilation of factor \(1 \over k\) from \(y\)-axis -\end{itemize} - -\begin{tikzpicture} - \begin{axis}[restrict x to domain=-0.9:0.9, axis y line = middle, yticklabels={,,}, xticklabels={,,}, enlargelimits, ticks=none] - \addplot[red, thick, smooth, samples=100] plot (\x, {pow(2,x)}) node[below, pos=1] {\(2^x\)}; - \addplot[blue, thick, smooth, samples=100] plot (\x, {pow(3,x)}) node[left, pos=1] {\(3^x\)}; - \addplot[orange, thick, smooth, samples=100] plot (\x, {pow(e,x)}) node[below, pos=1] {\(e^x\)}; - \addplot[mark=*] coordinates {(0,1)} node[above left]{\((0,1)\)} ; - \addplot[purple, ultra thick, dashed] plot (\x, 0) node[black, below, font=\footnotesize, pos=0.75] {\(y=0\)}; - \end{axis} -\end{tikzpicture} - -\subsection*{Graphing logarithmic functions} - -\(\log_e x\) is the inverse of \(e^x\) (reflection across \(y=x\)) - -\[f(x)=A \log_a k(x-b) + c\] - -where - -\begin{itemize} -\tightlist -\item - \textbf{domain} is \((b, \infty)\) -\item - \textbf{range} is \(\mathbb{R}\) -\item - \textbf{vertical asymptote} at \(x=b\) -\item - \(y\)-intercept exists if \(b<0\) -\item - dilation of factor \(|A|\) from \(x\)-axis -\item - dilation of factor \(1 \over k\) from \(y\)-axis -\end{itemize} -\begin{tikzpicture} - \begin{axis}[axis lines=middle, xmin=-0.5, xmax=5, ymin=-2, ymax=3, ticks=none] - \addplot[purple, ultra thick, dashed] coordinates {(0,-1.8) (0,2.8)} node[black, below right, pos=0.75, font=\footnotesize] {\(x=0\)}; - \addplot[orange,thick,domain=0.01:4,smooth,samples=100] {ln(x)} node[right, pos=1] {\(\log_e x\)}; - \addplot[red,thick,domain=0.01:4,smooth,samples=100] {log2(x)} node[right, pos=1] {\(\log_2 x\)}; - \addplot[blue,thick,domain=0.01:4,smooth,samples=100] {ln(x)/ln(3)} node[below right, pos=1] {\(\log_3 x\)}; - \addplot[mark=*] coordinates {(1,0)} node[above left]{\((0,1)\)} ; - \end{axis} -\end{tikzpicture} - -\subsection*{Finding equations} - -\colorbox{cas}{On CAS:} -\includegraphics[width=0.78125in]{graphics/cas-simultaneous.png} diff --git a/methods/temp/transformations.tex b/methods/temp/transformations.tex deleted file mode 100644 index d2b253c..0000000 --- a/methods/temp/transformations.tex +++ /dev/null @@ -1,168 +0,0 @@ -\definecolor{shade1}{HTML}{ffffff} -\definecolor{shade2}{HTML}{e6f2ff} -\definecolor{shade3}{HTML}{cce2ff} -\section{Transformations} - -\textbf{Order of operations:} DRT - -\begin{center}dilations --- reflections --- translations\end{center} - -\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)} - -\begin{itemize} -\tightlist -\item - dilation factor of \(|a|\) units parallel to \(y\)-axis or from - \(x\)-axis -\item - if \(a<0\), graph is reflected over \(x\)-axis -\item - translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis -\item - translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis -\item - for \((ax)^n\), dilation factor is \(1 \over a\) parallel to - \(x\)-axis or from \(y\)-axis -\item - when \(0 < |a| < 1\), graph becomes closer to axis -\end{itemize} - -\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)} - -Applies to exponential, log, trig, \(e^x\), polynomials.\\ -Functions must be written in form \(y=Af[n(x+c)]+b\) - -\begin{itemize} -\tightlist -\item - dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection - across \(y\)-axis) -\item - dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\), - reflection across \(x\)-axis) -\item - translation of \(c\) units from \(y\)-axis (\(x\)-shift) -\item - translation of \(b\) units from \(x\)-axis (\(y\)-shift) -\end{itemize} - -\subsection*{Dilations} - -Two pairs of equivalent processes for \(y=f(x)\): - -\begin{enumerate} -\def\labelenumi{\arabic{enumi}.} -\item - \begin{itemize} - \tightlist - \item - Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\) - \item - Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\) - \end{itemize} -\item - \begin{itemize} - \tightlist - \item - Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\) - \item - Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\) - \end{itemize} -\end{enumerate} - -For graph of \(y={1 \over x}\), horizontal \& vertical dilations are -equivalent (symmetrical). If \(y={a \over x}\), graph is contracted -rather than dilated. - -\subsection*{Matrix transformations} - -Find new point \((x^\prime, y^\prime)\). Substitute these into original -equation to find image with original variables \((x, y)\). - -\subsection*{Reflections} - -\begin{itemize} -\tightlist -\item - Reflection \textbf{in} axis = reflection \textbf{over} axis = - reflection \textbf{across} axis -\item - Translations do not change -\end{itemize} - -\subsection*{Translations} - -For \(y = f(x)\), these processes are equivalent: - -\begin{itemize} -\tightlist -\item - applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the - graph of \(y = f(x)\) -\item - replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain - \(y-k = f(x-h)\) -\end{itemize} - -\subsection*{Power functions} - -\textbf{Strictly increasing:} \(f(x_2) > f(x_1)\) where \(x_2 > x_1\) -(including \(x=0\)) - - -\subsubsection*{\(x^{-1 \over n}\) where \(n \in \mathbb{Z}^+\)} - -Mostly only on CAS. - -We can write -\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\ -Domain is: -\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\) - -If \(n\) is odd, it is an odd function. - -\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)} - -\[x^{p \over q} = \sqrt[q]{x^p}\] - -\begin{itemize} -\tightlist -\item - if \(p > q\), the shape of \(x^p\) is dominant -\item - if \(p < q\), the shape of \(x^{1 \over q}\) is dominant -\item - points \((0, 0)\) and \((1, 1)\) will always lie on graph -\item - Domain is: - \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\) -\end{itemize} - -\subsection*{Piecewise functions} - -\[\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}\] - -\textbf{Open circle:} point included\\ -\textbf{Closed circle:} point not included - -\subsection*{Operations on functions} - -For \(f \pm g\) and \(f \times g\): -\quad \(\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)\) - -Addition of linear piecewise graphs: add \(y\)-values at key points - -Product functions: - -\begin{itemize} -\tightlist -\item - product will equal 0 if \(f=0\) or \(g=0\) -\item - \(f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0\) -\end{itemize} - -\subsection*{Composite functions} - -\((f \circ g)(x)\) is defined iff -\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\) diff --git a/methods/transformations.tex b/methods/transformations.tex new file mode 100644 index 0000000..00428e3 --- /dev/null +++ b/methods/transformations.tex @@ -0,0 +1,134 @@ +\definecolor{shade1}{HTML}{ffffff} +\definecolor{shade2}{HTML}{e6f2ff} +\definecolor{shade3}{HTML}{cce2ff} +\section{Transformations} + +\textbf{Order of operations:} DRT + +\begin{center}dilations --- reflections --- translations\end{center} + +\subsection*{Transforming \(x^n\) to \(a(x-h)^n+K\)} + +\begin{itemize} +\tightlist +\item + dilation factor of \(|a|\) units parallel to \(y\)-axis or from + \(x\)-axis +\item + if \(a<0\), graph is reflected over \(x\)-axis +\item + translation of \(k\) units parallel to \(y\)-axis or from \(x\)-axis +\item + translation of \(h\) units parallel to \(x\)-axis or from \(y\)-axis +\item + for \((ax)^n\), dilation factor is \(1 \over a\) parallel to + \(x\)-axis or from \(y\)-axis +\item + when \(0 < |a| < 1\), graph becomes closer to axis +\end{itemize} + +\subsection*{Transforming \(f(x)\) to \(y=Af[n(x+c)]+b\)} + +Applies to exponential, log, trig, \(e^x\), polynomials.\\ +Functions must be written in form \(y=Af[n(x+c)]+b\) + +\begin{itemize} +\tightlist +\item + dilation by factor \(|A|\) from \(x\)-axis (if \(A<0\), reflection + across \(y\)-axis) +\item + dilation by factor \(1 \over n\) from \(y\)-axis (if \(n<0\), + reflection across \(x\)-axis) +\item + translation of \(c\) units from \(y\)-axis (\(x\)-shift) +\item + translation of \(b\) units from \(x\)-axis (\(y\)-shift) +\end{itemize} + +\subsection*{Dilations} + +Two pairs of equivalent processes for \(y=f(x)\): + +\begin{enumerate} +\def\labelenumi{\arabic{enumi}.} +\item + \begin{itemize} + \tightlist + \item + Dilating from \(x\)-axis: \((x, y) \rightarrow (x, by)\) + \item + Replacing \(y\) with \(y \over b\) to obtain \(y = b f(x)\) + \end{itemize} +\item + \begin{itemize} + \tightlist + \item + Dilating from \(y\)-axis: \((x, y) \rightarrow (ax, y)\) + \item + Replacing \(x\) with \(x \over a\) to obtain \(y = f({x \over a})\) + \end{itemize} +\end{enumerate} + +For graph of \(y={1 \over x}\), horizontal \& vertical dilations are +equivalent (symmetrical). If \(y={a \over x}\), graph is contracted +rather than dilated. + +\subsection*{Matrix transformations} + +Find new point \((x^\prime, y^\prime)\). Substitute these into original +equation to find image with original variables \((x, y)\). + +\subsection*{Reflections} + +\begin{itemize} +\tightlist +\item + Reflection \textbf{in} axis = reflection \textbf{over} axis = + reflection \textbf{across} axis +\item + Translations do not change +\end{itemize} + +\subsection*{Translations} + +For \(y = f(x)\), these processes are equivalent: + +\begin{itemize} +\tightlist +\item + applying the translation \((x, y) \rightarrow (x + h, y + k)\) to the + graph of \(y = f(x)\) +\item + replacing \(x\) with \(x-h\) and \(y\) with \(y-k\) to obtain + \(y-k = f(x-h)\) +\end{itemize} + +\subsection*{Power functions} + +Mostly only on CAS. + +We can write +\(x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}\)n.\\ +Domain is: +\(\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}\) + +If \(n\) is odd, it is an odd function. + +\subsubsection*{\(x^{p \over q}\) where \(p, q \in \mathbb{Z}^+\)} + +\[x^{p \over q} = \sqrt[q]{x^p}\] + +\begin{itemize} +\tightlist +\item + if \(p > q\), the shape of \(x^p\) is dominant +\item + if \(p < q\), the shape of \(x^{1 \over q}\) is dominant +\item + points \((0, 0)\) and \((1, 1)\) will always lie on graph +\item + Domain is: + \(\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}\) +\end{itemize} +