--- /dev/null
+\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}
+
\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}
\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}
\usepackage{amssymb}
\usepackage{harpoon}
\usepackage{tabularx}
+\usepackage{tabu}
\usepackage{makecell}
\usepackage[dvipsnames, table]{xcolor}
\usepackage{blindtext}
\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}
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}
--- /dev/null
+\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}
+++ /dev/null
-\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}
-
+++ /dev/null
-\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}
+++ /dev/null
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- hidelinks,
- pdfcreator={LaTeX via pandoc}}
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-\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}
+++ /dev/null
-\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}
+++ /dev/null
-\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
-\(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)
--- /dev/null
+\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}
+