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\fancyhead[LO,LE]{Year 12 Methods}
\fancyhead[CO,CE]{Andrew Lorimer}
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\begin{document}

\title{\vspace{-20mm}Year 12 Methods}
\author{Andrew Lorimer}
\date{}
\maketitle


\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
          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
      \(\operatorname{ran}(g) \subseteq \operatorname{dom}(f)\)

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      \begin{figure*}[ht]
        \centering

        \begin{tabularx}{\textwidth}{r|Y|Y}

          & \(n\) is even & \(n\) is odd \\ \hline

          \centering \(x^n, n \in \mathbb{Z}^+\) & 

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            \begin{axis}[blank, xmin=-3,  xmax=3]
              \addplot[blankplot] {(x^2)};
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          \end{tikzpicture}} \\ \hline

          \centering \(x^n, n \in \mathbb{Z}^-\) &

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            \begin{axis}[blank, xmin=-4, xmax=4, ymax=8, ymin=-0]
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          \end{tikzpicture}} &

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              \addplot[blankplot, domain=0.1:3] {(x^(-1))};
            \end{axis}
          \end{tikzpicture}} \\ \hline

          \centering \(x^{\frac{1}{n}}, n \in \mathbb{Z}^-\) &

          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
            \begin{axis}[blank, xmin=-1,  xmax=5]
              \addplot[blankplot] {(x^(1/2))};
            \end{axis}
          \end{tikzpicture}} &

          \adjustbox{margin=0 1ex, valign=m}{\begin{tikzpicture}
            \begin{axis}[blank, xmin=-3, xmax=3, ymin=-3, ymax=3]
              \addplot [blankplot, domain=-2:2] gnuplot[id=poly]{sgn(x)*(abs(x)**(1./3)) };
            \end{axis}
          \end{tikzpicture}} \\ \hline

        \end{tabularx}
      \end{figure*}

      \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}



      \section{Statistics}

      \subsection*{Probability}

      \begin{align*}
        \Pr(A \cup B) &= \Pr(A) + \Pr(B) - \Pr(A \cap B) \\
        \Pr(A \cap B) &= \Pr(A|B) \times \Pr(B) \\
        \Pr(A|B) &= \frac{\Pr(A \cap B)}{\Pr(B)} \\
        \Pr(A) &= \Pr(A|B) \cdot \Pr(B) + \Pr(A|B^{\prime}) \cdot \Pr(B^{\prime})
      \end{align*}

      Mutually exclusive \(\implies \Pr(A \cup B) = 0\) \\

      Independent events:
      \begin{flalign*}
        \quad \Pr(A \cap B) &= \Pr(A) \times \Pr(B)& \\
        \Pr(A|B) &= \Pr(A) \\
        \Pr(B|A) &= \Pr(B)
      \end{flalign*}

      \subsection*{Combinatorics}

      \begin{itemize}
        \item Arrangements \({n \choose k} = \frac{n!}{(n-k)}\)
        \item \colorbox{important}{Combinations} \({n \choose k} = \frac{n!}{k!(n-k)!}\)
        \item Note \({n \choose k} = {n \choose k-1}\)
      \end{itemize}

      \subsection*{Distributions}

      \subsubsection*{Mean \(\mu\)}

      \textbf{Mean} \(\mu\) or \textbf{expected value} \(E(X)\)

      \begin{align*}
        E(X) &= \frac{\Sigma \left[ x \cdot f(x) \right]}{\Sigma f} \tag{\(f =\) absolute frequency} \\
        &= \sum_{i=1}^n \left[ x_i \cdot \Pr(X=x_i) \right] \tag{discrete}\\
        &= \int_\textbf{X} (x \cdot f(x)) \> dx
      \end{align*}

      \subsubsection*{Mode}

      Most popular value (has highest probability of all \(X\) values). Multiple modes can exist if \(>1 \> X\) value have equal-highest probability. Number must exist in distribution.

      \subsubsection*{Median}

      If \(m > 0.5\), then value of \(X\) that is reached is the median of \(X\). If \(m = 0.5 = 0.5\), then \(m\) is halfway between this value and the next. To find \(m\), add values of \(X\) from smallest to alrgest until the sum reaches 0.5.

      \[ m = X \> \text{such that} \> \int_{-\infty}^{m} f(x) dx = 0.5 \]

      \subsubsection*{Variance \(\sigma^2\)}

      \begin{align*}
        \operatorname{Var}(x) &= \sum_{i=1}^n p_i (x_i-\mu)^2 \\
        &= \sum (x-\mu)^2 \times \Pr(X=x) \\
        &= \sum x^2 \times p(x) - \mu^2 \\
        &= \operatorname{E}(X^2) - [\operatorname{E}(X)]^2
        &= E\left[(X-\mu)^2\right]
      \end{align*}

      \subsubsection*{Standard deviation \(\sigma\)}

      \begin{align*}
        \sigma &= \operatorname{sd}(X) \\
        &= \sqrt{\operatorname{Var}(X)}
      \end{align*}

      \subsection*{Binomial distributions}

      Conditions for a \textit{binomial distribution}:
      \begin{enumerate}
        \item Two possible outcomes: \textbf{success} or \textbf{failure}
        \item \(\Pr(\text{success})\) is constant across trials (also denoted \(p\))
        \item Finite number \(n\) of independent trials
      \end{enumerate}


      \subsubsection*{Properties of \(X \sim \operatorname{Bi}(n,p)\)}

      \begin{align*}
        \mu(X) &= np \\
        \operatorname{Var}(X) &= np(1-p) \\
        \sigma(X) &= \sqrt{np(1-p)} \\
        \Pr(X=x) &= {n \choose x} \cdot p^x \cdot (1-p)^{n-x}
      \end{align*}

      \begin{cas}
        Interactive \(\rightarrow\) Distribution \(\rightarrow\) \verb;binomialPdf; then input
        \begin{description}[nosep, style=multiline, labelindent=0.5cm, leftmargin=3cm, font=\normalfont]
          \item [x:] no. of successes
          \item [numtrial:] no. of trials
          \item [pos:] probability of success
        \end{description}
      \end{cas}

      \subsection*{Continuous random variables}

      A continuous random variable \(X\) has a pdf \(f\) such that:

      \begin{enumerate}
        \item \(f(x) \ge 0 \forall x \)
        \item \(\int^\infty_{-\infty} f(x) \> dx = 1\)
      \end{enumerate}

      \begin{align*}
        E(X) &= \int_\textbf{X} (x \cdot f(x)) \> dx \\
        \operatorname{Var}(X) &= E\left[(X-\mu)^2\right]
      \end{align*}

      \[ \Pr(X \le c) = \int^c_{-\infty} f(x) \> dx \]


      \subsection*{Two random variables \(X, Y\)}

      If \(X\) and \(Y\) are independent:
      \begin{align*}
        \operatorname{E}(aX+bY) & = a\operatorname{E}(X)+b\operatorname{E}(Y) \\
        \operatorname{Var}(aX \pm bY \pm c) &= a^2 \operatorname{Var}(X) + b^2 \operatorname{Var}(Y)
      \end{align*}

      \subsection*{Linear functions \(X \rightarrow aX+b\)}

      \begin{align*}
        \Pr(Y \le y) &= \Pr(aX+b \le y) \\
        &= \Pr\left(X \le \dfrac{y-b}{a}\right) \\
        &= \int^{\frac{y-b}{a}}_{-\infty} f(x) \> dx
      \end{align*}

      \begin{align*}
        \textbf{Mean:} && \operatorname{E}(aX+b) & = a\operatorname{E}(X)+b \\
        \textbf{Variance:} && \operatorname{Var}(aX+b) &= a^2 \operatorname{Var}(X) \\
      \end{align*}

      \subsection*{Expectation theorems}

      For some non-linear function \(g\), the expected value \(E(g(X))\) is not equal to \(g(E(X))\).

      \begin{align*}
        E(X^2) &= \operatorname{Var}(X) - \left[E(X)\right]^2 \\
        E(X^n) &= \Sigma x^n \cdot p(x) \tag{non-linear} \\
        &\ne [E(X)]^n \\
        E(aX \pm b) &= aE(X) \pm b \tag{linear} \\
        E(b) &= b \tag{\(\forall b \in \mathbb{R}\)}\\
        E(X+Y) &= E(X) + E(Y) \tag{two variables}
      \end{align*}

      \subsection*{Sample mean}

      Approximation of the \textbf{population mean} determined experimentally.

      \[ \overline{x} = \dfrac{\Sigma x}{n} \]

      where
      \begin{description}[nosep, labelindent=0.5cm]
        \item \(n\) is the size of the sample (number of sample points)
        \item \(x\) is the value of a sample point
      \end{description}

      \begin{cas}
        \begin{enumerate}[leftmargin=3mm]
          \item Spreadsheet
          \item In cell A1:\\ \path{mean(randNorm(sd, mean, sample size))}
          \item Edit \(\rightarrow\) Fill \(\rightarrow\) Fill Range
          \item Input range as A1:An where \(n\) is the number of samples
          \item Graph \(\rightarrow\) Histogram
        \end{enumerate}
      \end{cas}

      \subsubsection*{Sample size of \(n\)}

      \[ \overline{X} = \sum_{i=1}^n \frac{x_i}{n} = \dfrac{\sum x}{n} \]

      Sample mean is distributed with mean \(\mu\) and sd \(\frac{\sigma}{\sqrt{n}}\) (approaches these values for increasing sample size \(n\)).

      For a new distribution with mean of \(n\) trials, \(\operatorname{E}(X^\prime) = \operatorname{E}(X), \quad \operatorname{sd}(X^\prime) = \dfrac{\operatorname{sd}(X)}{\sqrt{n}}\)

      \begin{cas}

        \begin{itemize}
          \item Spreadsheet \(\rightarrow\) Catalog \(\rightarrow\) \verb;randNorm(sd, mean, n); where \verb;n; is the number of samples. Show histogram with Histogram key in top left
          \item To calculate parameters of a dataset: Calc \(\rightarrow\) One-variable
        \end{itemize}

      \end{cas}

      \subsection*{Normal distributions}


      \[ Z = \frac{X - \mu}{\sigma} \]

      Normal distributions must have area (total prob.) of 1 \(\implies \int^\infty_{-\infty} f(x) \> dx = 1\) \\
      \(\text{mean} = \text{mode} = \text{median}\)

      \begin{warning}
        Always express \(z\) as +ve. Express confidence \textit{interval} as ordered pair.
      \end{warning}

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                  ylabel={\(\Pr(X=x)\)}]
                  \addplot {gauss(0,0.75)};
                  \fill[red!30] (-3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-((x)^2)/(2*0.75^2))} -- (3,0) -- cycle;
                  \fill[darkgray!30] (3,0)  -- plot[id=f3,domain=-3:3,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (3,0) -- cycle;
                  \fill[lightgray!30] (-2,0)  -- plot[id=f3,domain=-2:2,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (2,0) -- cycle;
                  \fill[white!30] (-1,0)  -- plot[id=f3,domain=-1:1,samples=50] function {1/(0.75*sqrt(2*pi))*exp(-x*x*0.5/(0.75*0.75))} -- (1,0) -- cycle;
                  \begin{scope}[<->]
                    \draw (-1,0.35) -- (1,0.35) node [midway, fill=white] {68.3\%};
                    \draw (-2,0.25) -- (2,0.25) node [midway, fill=white] {95.5\%};
                    \draw (-3,0.15) -- (3,0.15) node [midway, fill=white] {99.7\%};
                  \end{scope}
                  \begin{scope}[-, dashed, gray]
                    \draw (-1,0) -- (-1, 0.35);
                    \draw (1,0) -- (1, 0.35);
                    \draw (-2,0) -- (-2, 0.25);
                    \draw (2,0) -- (2, 0.25);
                    \draw (-3,0) -- (-3, 0.15);
                    \draw (3,0) -- (3, 0.15);
                  \end{scope}
                \end{axis}
                \begin{axis}[every axis plot post/.append style={
                    mark=none,domain=-3:3,samples=50,smooth}, 
                  axis x line=bottom, 
                  enlargelimits=upper,
                  x=\textwidth/10,
                  xtick={-2,-1,0,1,2},
                  axis x line shift=30pt,
                  hide y axis,
                  x tick label style = {font=\footnotesize},
                  xlabel={\(Z\)},
                  every axis x label/.style={at={(axis description cs:1,-0.25)},anchor=south west}]
                  \addplot {gauss(0,0.75)};
                \end{axis}
              \end{tikzpicture}
            \end{figure*}
          \end{document}
