[methods] fix formatting of notes

diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md
index 215845dc033695058ee7b9f03272b801100faf46..e12b37e76611a8ec1714f7877ef26ab7f979510e 100644 (file)
--- a/methods/calculus-ref.md
+++ b/methods/calculus-ref.md
@@ -1,25 +1,29 @@
 ---
 ---

-geometry: margin=1cm
+geometry: a4paper, margin=2cm

 columns: 2
 columns: 2

-graphics: yes
-tables: yes

 author: Andrew Lorimer
 header-includes:
 author: Andrew Lorimer
 header-includes:

+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}

 - \usepackage{tabularx}
 - \usepackage{tabularx}

+- \usepackage[dvipsnames, table]{xcolor}

 ---
 
 ---
 

-
+\linespread{3}

 \pagenumbering{gobble}
 \renewcommand{\arraystretch}{1.4}
 \pagenumbering{gobble}
 \renewcommand{\arraystretch}{1.4}

+\definecolor{cas}{HTML}{e6f0fe}

 
 

-
-# Methods - Calculus
+# Calculus

 
 ## Average rate of change
 
 $$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
 
 
 ## Average rate of change
 
 $$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$
 

-On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$
+\colorbox{cas}{On CAS:} Action $\rightarrow$ Calculation $\rightarrow$ `diff`

 
 ## Instantaneous rate of change
 
 
 ## Instantaneous rate of change
 


@@ -51,18 +55,22 @@ Not differentiable at:
 **Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
 **Secant** $={{f(x+h)-f(x)} \over h}$
 
 **Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$)  
 **Secant** $={{f(x+h)-f(x)} \over h}$
 

-## Strictly increasing
+## Strictly increasing/decreasing

 
 

-- **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$
-- **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$
-- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing**
-- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing**
+For $x_2$ and $x_1$ where $x_2 > x_1$:
+
+- **strictly increasing** where $f(x_2) > f(x_1)$  
+or $f^\prime(x)>0$
+- **strictly decreasing** where $f(x_2) < f(x_1)$  
+or $f^\prime(x)<0$

 - Endpoints are included, even where gradient $=0$
 
 - Endpoints are included, even where gradient $=0$
 

+\columnbreak
+

 ### Solving on CAS
 
 ### Solving on CAS
 

-**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
-**In graph**: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.
+\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line)  
+\colorbox{cas}{\textbf{In graph}}: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line.

 
 ## Stationary points
 
 
 ## Stationary points
 


@@ -87,11 +95,14 @@ Find derivative, solve for ${dy \over dx} = 0$
 
 ## Function derivatives
 
 
 ## Function derivatives
 

-\begin{tabularx}{\columnwidth}{rl}
+\definecolor{shade1}{HTML}{ffffff}
+\definecolor{shade2}{HTML}{F0F9E4}
+\rowcolors{1}{shade1}{shade2}
+\begin{tabularx}{\columnwidth}{rX}

   
   \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
 
   
   \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline
 

-  \(kx^n\) & \(knx^{n-1}\)\tabularnewline
+  \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline

   \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
   \(c\) & \(0\)\tabularnewline
   \({u \over v}\) &
   \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline
   \(c\) & \(0\)\tabularnewline
   \({u \over v}\) &

diff --git a/methods/calculus-ref.pdf b/methods/calculus-ref.pdf
index 06fb92880bc68c5d098cbadd65499cc2c61f7e45..fa940f3b812ad341235efe0cef2e4f2921d35634 100644 (file)
Binary files a/methods/calculus-ref.pdf and b/methods/calculus-ref.pdf differ

diff --git a/methods/circ-functions.md b/methods/circ-functions.md
index e639e709de0772c90de4be0cc9ba261c25289b83..27ce6d5201c997f66f667639565274cf24580b19 100644 (file)
--- a/methods/circ-functions.md
+++ b/methods/circ-functions.md
@@ -1,21 +1,27 @@
 ---
 ---

-geometry: margin=2cm
+geometry: a4paper, margin=2cm

 columns: 2
 columns: 2

-graphics: yes
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \usepackage{graphicx}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}

 ---
 ---

-# Circular functions

 
 

-<!-- ## Radians and degrees -->
+\setstretch{1.2}
+\pagenumbering{gobble}

 
 

-<!-- $$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$ -->
+# Circular functions

 
 ## Exact values
 
 \includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
 \includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
 
 
 ## Exact values
 
 \includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png}
 \includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png}
 

-<!-- ![diag](graphics/exact-values-1.png) -->
-<!-- ![diag](graphics/exact-values-2.png) -->
+$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$

 
 ## $\sin$ and $\cos$ graphs
 
 
 ## $\sin$ and $\cos$ graphs
 

diff --git a/methods/circ-functions.pdf b/methods/circ-functions.pdf
index 84f3121b4646c395aa13270f7608f013ed928ae3..990d0dc970c086eb19a5ebf1c1318866192ad7c3 100644 (file)
Binary files a/methods/circ-functions.pdf and b/methods/circ-functions.pdf differ

diff --git a/methods/inverse-functions.md b/methods/inverse-functions.md
index 6710e9c8cdae90ff827269e866d5750cc23aa43f..8b4de947cb5835cdcef1027d39a3400346251c08 100644 (file)
--- a/methods/inverse-functions.md
+++ b/methods/inverse-functions.md
@@ -1,3 +1,17 @@
+---
+geometry: margin=2cm
+author: Andrew Lorimer
+header-includes:
+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+---
+
+\setstretch{1.3}
+\pagenumbering{gobble}
+

 # Inverse functions
 
 ## Functions
 # Inverse functions
 
 ## Functions


@@ -7,24 +21,24 @@
 
 ## One to one functions
 
 
 ## One to one functions
 

-- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$
-- i.e. unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is)
+- $f(x)$ is *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)

 - horizontal line test
 - if not one to one, it is many to one
 
 - horizontal line test
 - if not one to one, it is many to one
 

-## Inverse functions $f^{-1}$
+## Deriving $f^{-1}$

 
 - if $f(g(x)) = x$, then $g$ is the inverse of $f$
 - reflection across $y-x$
 - $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$
 
 - if $f(g(x)) = x$, then $g$ is the inverse of $f$
 - reflection across $y-x$
 - $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$

-- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)
-- - $\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one
+- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test)  
+$\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one

 - $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$
 
 - $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$
 

-Requirements for showing working for $f^{-1}$:
+### Requirements for showing working for $f^{-1}$

 
 

-- start with *"let $y=f(x)$"*
-- must state *"take inverse"* for line where $y$ and $x$ are swapped
-- do all working in terms of $y=\dots$
-- for square root, state $\pm$ solutions then show restricted
-- for inverse *function*, state in function notation
+1. start with *"let $y=f(x)$"*
+2. must state *"take inverse"* for line where $y$ and $x$ are swapped
+3. do all working in terms of $y=\dots$
+4. for square root, state $\pm$ solutions then show restricted
+5. for inverse *function*, state in function notation

diff --git a/methods/inverse-functions.pdf b/methods/inverse-functions.pdf
index 5e5f1f853febb16d19229df0cc83f9b55181fe0e..a5e91c489b5f10283522d9b2e423168a75029492 100644 (file)
Binary files a/methods/inverse-functions.pdf and b/methods/inverse-functions.pdf differ

diff --git a/methods/polynomials-ref.pdf b/methods/polynomials-ref.pdf
index 98b6422296babb69dd107a7a60ce0befa8e97d26..48408124f50295fd2a9030969dd86825f8de0f46 100644 (file)
Binary files a/methods/polynomials-ref.pdf and b/methods/polynomials-ref.pdf differ

diff --git a/methods/polynomials.md b/methods/polynomials.md
index c826306003a45111c78051b0c8446e844a2e2727..6a4cf980115eb6f4c87e04a4ba7a8e4880a3f028 100644 (file)
--- a/methods/polynomials.md
+++ b/methods/polynomials.md
@@ -1,10 +1,20 @@
 ---
 ---

-geometry: margin=1.5cm
+geometry: a4paper, margin=2cm

 columns: 2
 columns: 2

+author: Andrew Lorimer

 header-includes:
 header-includes:

+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}

 - \usepackage{tabularx}
 - \usepackage{tabularx}

+- \usepackage[dvipsnames]{xcolor}

 ---
 
 ---
 

+\setstretch{1.3}
+\definecolor{cas}{HTML}{e6f0fe}

 \pagenumbering{gobble}
 \renewcommand{\arraystretch}{1.4}
 
 \pagenumbering{gobble}
 \renewcommand{\arraystretch}{1.4}
 


@@ -13,12 +23,15 @@ header-includes:
 ## Quadratics
 
 \newcolumntype{R}{>{\raggedleft\arraybackslash}X}
 ## Quadratics
 
 \newcolumntype{R}{>{\raggedleft\arraybackslash}X}

-\begin{tabularx}{\columnwidth}{|R|l|}
-  Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\
-  & where $mn=c, \> m+n=b$ \\
+\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 & $a^2 - b^2 = (a - b)(a + b)$ \\
   Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\

-  Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\
+  \hline
+  Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\
+  \hline

   Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\
   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}
 
   Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\
 \end{tabularx}
 


@@ -28,6 +41,13 @@ header-includes:
 **Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
 **Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
 
 **Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$  
 **Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$  
 

+$$y=a(bx-h)^3 + c$$
+
+- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
+- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
+- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
+- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
+

 ## Linear and quadratic graphs
 
 ### Forms of linear equations
 ## Linear and quadratic graphs
 
 ### Forms of linear equations


@@ -40,17 +60,7 @@ $y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts
 
 Parallel lines: $m_1 = m_2$  
 Perpendicular lines: $m_1 \times m_2 = -1$  
 
 Parallel lines: $m_1 = m_2$  
 Perpendicular lines: $m_1 \times m_2 = -1$  

-Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$
-
-
-## Cubic graphs
-
-$$y=a(bx-h)^3 + c$$
-
-- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$)
-- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$
-- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$
-- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$
+Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$

 
 ## Quartic graphs
 
 
 ## Quartic graphs
 


@@ -62,17 +72,13 @@ $y=ax^2(x-b)(x-c)$
 $y=a(x-b)^2(x-c)^2$  
 $y=a(x-b)(x-c)^3$
 
 $y=a(x-b)^2(x-c)^2$  
 $y=a(x-b)(x-c)^3$
 

-## Literal equations
-
-Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters)
-

 ## Simultaneous equations (linear)
 
 - **Unique solution** - lines intersect at point
 - **Infinitely many solutions** - lines are equal
 - **No solution** - lines are parallel
 
 ## Simultaneous equations (linear)
 
 - **Unique solution** - lines intersect at point
 - **Infinitely many solutions** - lines are equal
 - **No solution** - lines are parallel
 

-### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions
+### 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
 
 
 where all coefficients are known except for one, and $a, b$ are known
 


@@ -87,10 +93,10 @@ where all coefficients are known except for one, and $a, b$ are known
    - *--- for infinite/no solutions: ---*
 5. Substitute variable into both original equations
 6. Rearrange equations so that LHS of each is the same
    - *--- for infinite/no solutions: ---*
 5. Substitute variable into both original equations
 6. Rearrange equations so that LHS of each is the same

-7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions)  
-   If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions)
+7. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns)  
+   $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns)

 
 

-Or use Matrix -> `det` on CAS.
+\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det`

 
 ### 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 \\
 
 ### 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 \\

diff --git a/methods/stuff.md b/methods/stuff.md
index 2055b63a7d86e0fb419e1f9b851304547ead4d5b..d48dadb9c714aa4e57a2991cbdbdb8c49a6f6906 100644 (file)
--- a/methods/stuff.md
+++ b/methods/stuff.md
@@ -1,79 +1,69 @@
 ---
 ---

-geometry: margin=1.5cm
-<!-- columns: 2 -->
-graphics: yes
-tables: yes
+geometry: a4paper, margin=2cm
+columns: 2

 author: Andrew Lorimer
 author: Andrew Lorimer

-classoption: twocolumn
-header-includes: \pagenumbering{gobble}
+header-includes:
+- \usepackage{fancyhdr}
+- \usepackage{setspace}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+- \usepackage[dvipsnames]{xcolor}

 ---
 
 ---
 

-# Exponential and Index Functions
+\pagenumbering{gobble}
+\setstretch{1.5}
+\definecolor{cas}{HTML}{e6f0fe}
+
+# Exponentials & Logarithms

 
 ## Index laws
 
 
 ## Index laws
 

-\begin{equation}\begin{split}
+\begin{equation*}\begin{split}

   a^m \times a^n & = a^{m+n} \\
   a^m \times a^n & = a^{m+n} \\

-  a^m \div a^n & = a^{m-n}4 \\
+  a^m \div a^n & = a^{m-n} \\

   (a^m)^n & = a^{_mn} \\
   (ab)^m & = a^m b^m \\
   (a^m)^n & = a^{_mn} \\
   (ab)^m & = a^m b^m \\

-  {({a \over b})}^m & = {a^m \over b^m}
-\end{split}\end{equation}
-
-## Fractional indices
-
-$$^n\sqrt{x}=x^{1/n}$$
-
-## Logarithms
-
-$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$
-
-## Using logs to solve index eq's
-
-Used for equations without common base exponent
-
-Or change base:  
-$$\log_b c = {{\log_a c} \over {\log_a b}}$$
-
-If $a<1, \quad \log_{b} a < 0$ (flip inequality operator)
-
-## Exponential functions
-
-$e^x$ - natural exponential function
-
-$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$
+  {({a \over b})}^m & = {a^m \over b^m} \\
+  ^n\sqrt{x} &=x^{1/n}
+\end{split}\end{equation*}

 
 ## Logarithm laws
 
 
 ## Logarithm laws
 

-\begin{equation}\begin{split}
+\begin{equation*}\begin{split}

   \log_a(mn) & = \log_am + \log_an \\
   \log_a({m \over n}) & = \log_am - \log_a \\
   \log_a(m^p) & = p\log_am \\
   \log_a(m^{-1}) & = -\log_am \\
   \log_a(mn) & = \log_am + \log_an \\
   \log_a({m \over n}) & = \log_am - \log_a \\
   \log_a(m^p) & = p\log_am \\
   \log_a(m^{-1}) & = -\log_am \\

-  \log_a1 = 0 & \text{ and } \log_aa = 1
-\end{split}\end{equation}
-
+  \log_a1 = 0 & \text{ and } \log_aa = 1 \\
+  \log_b c &= {{\log_a c} \over {\log_a b}}
+\end{split}\end{equation*}

 
 ## Inverse functions
 
 For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
 
 
 ## Inverse functions
 
 For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is:
 

-$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$
+$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$
+
+## Exponentials

 
 

-## Euler's number
+$$e^x \quad \text{natural exponential function}$$

 
 $$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
 
 
 $$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$
 

-## Exponential and logarithmic modelling
+## Modelling

 
 $$A = A_0 e^{kt}$$
 
 
 $$A = A_0 e^{kt}$$
 

-where  
-$A_0$ is initial value  
-$t$ is time taken  
-$k$ is a constant  
-For continuous growth, $k > 0$  
-For continuous decay, $k < 0$
+- $A_0$ is initial value
+- $t$ is time taken
+- $k$ is a constant
+- For continuous growth, $k > 0$
+- For continuous decay, $k < 0$
+
+\columnbreak

 
 ## Graphing exponential functions
 
 
 ## Graphing exponential functions
 


@@ -83,7 +73,7 @@ $$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$
 - **horizontal asymptote** at $y=c$
 - **domain** is $\mathbb{R}$
 - **range** is $(c, \infty)$
 - **horizontal asymptote** at $y=c$
 - **domain** is $\mathbb{R}$
 - **range** is $(c, \infty)$

-- dilation of factor $A$ from $x$-axis
+- dilation of factor $|A|$ from $x$-axis

 - dilation of factor $1 \over k$ from $y$-axis
 
 ![](graphics/exponential-graphs.png){#id .class width=30%} 
 - dilation of factor $1 \over k$ from $y$-axis
 
 ![](graphics/exponential-graphs.png){#id .class width=30%} 


@@ -100,11 +90,11 @@ where
 - **range** is $\mathbb{R}$
 - **vertical asymptote** at $x=b$
 - $y$-intercept exists if $b<0$
 - **range** is $\mathbb{R}$
 - **vertical asymptote** at $x=b$
 - $y$-intercept exists if $b<0$

-- dilation of factor $A$ from $x$-axis
+- dilation of factor $|A|$ from $x$-axis

 - dilation of factor $1 \over k$ from $y$-axis
 
 ![](graphics/log-graphs.png){#id .class width=30%} 
 
 ## Finding equations
 
 - dilation of factor $1 \over k$ from $y$-axis
 
 ![](graphics/log-graphs.png){#id .class width=30%} 
 
 ## Finding equations
 

-Solve simultaneous equations on CAS: ![](graphics/cas-simultaneous.png){#id .class width=75px}
+\colorbox{cas}{On CAS:} ![](graphics/cas-simultaneous.png){#id .class width=75px}

diff --git a/methods/stuff.pdf b/methods/stuff.pdf
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diff --git a/methods/transformations-ref.pdf b/methods/transformations-ref.pdf
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diff --git a/methods/transformations.md b/methods/transformations.md
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--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -1,28 +1,48 @@
 ---
 ---

-geometry: margin=2cm
+geometry: a4paper, margin=2cm

 columns: 2
 author: Andrew Lorimer
 header-includes:
 columns: 2
 author: Andrew Lorimer
 header-includes:

+- \usepackage{setspace}
+- \usepackage{fancyhdr}
+- \pagestyle{fancy}
+- \fancyhead[LO,LE]{Year 12 Methods}
+- \fancyhead[CO,CE]{Andrew Lorimer}

 - \usepackage{graphicx}
 - \usepackage{tabularx}
 ---
 
 - \usepackage{graphicx}
 - \usepackage{tabularx}
 ---
 

-# Transformation
+\setstretch{1.6}
+\pagenumbering{gobble}

 
 

-**Order of operations:** DRT - Dilations, Reflections, Translations
+# Transformations
+
+**Order of operations:** DRT
+
+\begin{center}dilations --- reflections --- translations\end{center}

 
 ## Transforming $x^n$ to $a(x-h)^n+K$
 
 
 ## Transforming $x^n$ to $a(x-h)^n+K$
 

-- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
+- dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis

 - if $a<0$, graph is reflected over $x$-axis
 - if $a<0$, graph is reflected over $x$-axis

-- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
-- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
+- translation of $k$ units parallel to $y$-axis or from $x$-axis
+- translation of $h$ units parallel to $x$-axis or from $y$-axis

 - for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
 - when $0 < |a| < 1$, graph becomes closer to axis
 
 - for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
 - when $0 < |a| < 1$, graph becomes closer to axis
 

+## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
+
+Applies to exponential, log, trig, $e^x$, polynomials.  
+Functions must be written in form $y=Af[n(x+c)]+b$
+
+- dilation by factor $|A|$ from $x$-axis (if $A<0$, reflection across $y$-axis)
+- dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
+- translation of $c$ units from $y$-axis ($x$-shift)
+- translation of $b$ units from $x$-axis ($y$-shift)
+

 ## Dilations
 
 ## Dilations
 

-For the graph of $y = f(x)$, there are two pairs of equivalent processes:
+Two pairs of equivalent processes for $y=f(x)$:

 
 1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
    - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
 
 1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
    - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$


@@ -32,6 +52,10 @@ For the graph of $y = f(x)$, there are two pairs of equivalent processes:
 
 For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
 
 
 For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
 

+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+

 ## Reflections
 
 - Reflection **in** axis = reflection **over** axis = reflection **across** axis
 ## Reflections
 
 - Reflection **in** axis = reflection **over** axis = reflection **across** axis


@@ -42,17 +66,7 @@ For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (sy
 For $y = f(x)$, these processes are equivalent:
 
 - applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
 For $y = f(x)$, these processes are equivalent:
 
 - applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$

-- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
-
-## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
-
-Applies to exponential, log, trig, power, polynomial functions.  
-Functions must be written in form $y=Af[n(x+c)] + b$
-
-$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)  
-$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)  
-$c$ - translation from $y$-axis ($x$-shift)  
-$b$ - translation from $x$-axis ($y$-shift)
+- replacing $x$ with $x-h$ and $y$ with $y-k$ to obtain $y-k = f(x-h)$

 
 ## Power functions
 
 
 ## Power functions
 


@@ -65,26 +79,17 @@ Odd when $-f(x) = f(-x)$
 Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
 Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
 
 Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
 Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
 

-### $x^n$ where $n \in \mathbb{Z}^+$
-
-\begin{tabularx}{\textwidth}{|c|c|}
-  \(n\) is even & \(n\) is odd\\
-  {\includegraphics[height=1cm]{graphics/parabola.png}} & {\includegraphics[height=1cm]{graphics/cubic.png}}
-\end{tabularx}
-
-### $x^n$ where $n \in \mathbb{Z}^-$
-
-\begin{tabularx}{\textwidth}{|c|c|}
-  \(n\) is even & \(n\) is odd\\
-  {\includegraphics[height=1cm]{graphics/truncus.png}} & {\includegraphics[height=1cm]{graphics/hyperbola.png}}
-\end{tabularx}
-
-### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$

 
 

-\begin{tabularx}{\textwidth}{|c|c|}
-  \(n\) is even & \(n\) is odd\\
-  {\includegraphics[height=1cm]{graphics/square-root-graph.png}} & {\includegraphics[height=1cm]{graphics/cube-root-graph.png}}
-\end{tabularx}
+\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
+\begin{center}
+\begin{tabular}{m{1.2cm}|C|C}
+  & $n$ is even & $n$ is odd \\
+  \hline
+  \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
+  \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
+  \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
+\end{tabular}
+\end{center}

 
 ### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
 
 
 ### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
 


@@ -95,6 +100,8 @@ Domain is:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n
 
 If $n$ is odd, it is an odd function.
 
 
 If $n$ is odd, it is an odd function.
 

+\columnbreak
+

 ### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
 
 $$x^{p \over q} = \sqrt[q]{x^p}$$
 ### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
 
 $$x^{p \over q} = \sqrt[q]{x^p}$$


@@ -104,31 +111,23 @@ $$x^{p \over q} = \sqrt[q]{x^p}$$
 - points $(0, 0)$ and $(1, 1)$ will always lie on graph
 - Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 
 - points $(0, 0)$ and $(1, 1)$ will always lie on graph
 - Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 

-
-## Combinations of functions (piecewise/hybrid)
+## Piecewise functions

 
 $$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 
 
 $$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 

-Open circle - point included  
-Closed circle - point not included  
+**Open circle:** point included  
+**Closed circle:** point not included

 
 

-### Sum, difference, product of functions
-\begin{tabularx}{\columnwidth}{X|X}
-  sum & $f+g$ & domain $= \text{dom}(f) \cap \text{dom}(g)$ \\
-  difference & $f-g$ or $g-f$ & domain $=\text{dom}(f) \cap \text{dom}(g)$ \\
-  product & $f \times g$ & domain $=\text{dom}(f) \cap \text{dom}(g)$
-\end{tabularx}
-  
-Addition of linear piecewise graphs - add $y$-values at key points
+## Operations on functions

 
 

-Product functions:
+For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$

 
 

-- product will equal 0 if one of the functions is equal to 0
-- turning point on one function does not equate to turning point on product
+Addition of linear piecewise graphs: add $y$-values at key points

 
 

-## Matrix transformations
+Product functions:

 
 

-Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+- product will equal 0 if $f=0$ or $g=0$
+- $f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0$

 
 ## Composite functions
 
 
 ## Composite functions