methods / transformations.mdon commit [chem] start organic reactions summary document (5447e6c)
   1---
   2geometry: a4paper, margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{setspace}
   7- \usepackage{fancyhdr}
   8- \pagestyle{fancy}
   9- \fancyhead[LO,LE]{Year 12 Methods}
  10- \fancyhead[CO,CE]{Andrew Lorimer}
  11- \usepackage{graphicx}
  12- \usepackage{tabularx}
  13---
  14
  15\setstretch{1.6}
  16\pagenumbering{gobble}
  17
  18# Transformations
  19
  20**Order of operations:** DRT
  21
  22\begin{center}dilations --- reflections --- translations\end{center}
  23
  24## Transforming $x^n$ to $a(x-h)^n+K$
  25
  26- dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
  27- if $a<0$, graph is reflected over $x$-axis
  28- translation of $k$ units parallel to $y$-axis or from $x$-axis
  29- translation of $h$ units parallel to $x$-axis or from $y$-axis
  30- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
  31- when $0 < |a| < 1$, graph becomes closer to axis
  32
  33## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
  34
  35Applies to exponential, log, trig, $e^x$, polynomials.  
  36Functions must be written in form $y=Af[n(x+c)]+b$
  37
  38- dilation by factor $|A|$ from $x$-axis (if $A<0$, reflection across $y$-axis)
  39- dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)
  40- translation of $c$ units from $y$-axis ($x$-shift)
  41- translation of $b$ units from $x$-axis ($y$-shift)
  42
  43## Dilations
  44
  45Two pairs of equivalent processes for $y=f(x)$:
  46
  471. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
  48   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
  49
  502. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
  51   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
  52
  53For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
  54
  55## Matrix transformations
  56
  57Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
  58
  59## Reflections
  60
  61- Reflection **in** axis = reflection **over** axis = reflection **across** axis
  62- Translations do not change
  63
  64## Translations
  65
  66For $y = f(x)$, these processes are equivalent:
  67
  68- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
  69- replacing $x$ with $x-h$ and $y$ with $y-k$ to obtain $y-k = f(x-h)$
  70
  71## Power functions
  72
  73**Strictly increasing:**  $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$)
  74
  75### Odd and even functions
  76Even when $f(x) = -f(x)$  
  77Odd when $-f(x) = f(-x)$
  78
  79Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
  80Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
  81
  82
  83\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
  84\begin{center}
  85\begin{tabular}{m{1.2cm}|C|C}
  86  & $n$ is even & $n$ is odd \\
  87  \hline
  88  \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
  89  \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
  90  \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}}\\
  91\end{tabular}
  92\end{center}
  93
  94### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
  95
  96Mostly only on CAS.
  97
  98We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n.  
  99Domain is:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
 100
 101If $n$ is odd, it is an odd function.
 102
 103\columnbreak
 104
 105### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
 106
 107$$x^{p \over q} = \sqrt[q]{x^p}$$
 108
 109- if $p > q$, the shape of $x^p$ is dominant
 110- if $p < q$, the shape of $x^{1 \over q}$ is dominant
 111- points $(0, 0)$ and $(1, 1)$ will always lie on graph
 112- Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 113
 114## Piecewise functions
 115
 116$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 117
 118**Open circle:** point included  
 119**Closed circle:** point not included
 120
 121## Operations on functions
 122
 123For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$
 124
 125Addition of linear piecewise graphs: add $y$-values at key points
 126
 127Product functions:
 128
 129- product will equal 0 if $f=0$ or $g=0$
 130- $f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0$
 131
 132## Composite functions
 133
 134$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
 135
 136