methods / transformations.mdon commit [methods] tidy up & render notes for SAC (150b93e)
   1---
   2geometry: margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5header-includes:
   6- \usepackage{graphicx}
   7- \usepackage{tabularx}
   8---
   9
  10# Transformation
  11
  12**Order of operations:** DRT - Dilations, Reflections, Translations
  13
  14## Transforming $x^n$ to $a(x-h)^n+K$
  15
  16- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
  17- if $a<0$, graph is reflected over $x$-axis
  18- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
  19- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
  20- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
  21- when $0 < |a| < 1$, graph becomes closer to axis
  22
  23## Dilations
  24
  25For the graph of $y = f(x)$, there are two pairs of equivalent processes:
  26
  271. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
  28   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
  29
  302. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
  31   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
  32
  33For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
  34
  35## Reflections
  36
  37- Reflection **in** axis = reflection **over** axis = reflection **across** axis
  38- Translations do not change
  39
  40## Translations
  41
  42For $y = f(x)$, these processes are equivalent:
  43
  44- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
  45- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
  46
  47## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
  48
  49Applies to exponential, log, trig, power, polynomial functions.  
  50Functions must be written in form $y=Af[n(x+c)] + b$
  51
  52$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)  
  53$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)  
  54$c$ - translation from $y$-axis ($x$-shift)  
  55$b$ - translation from $x$-axis ($y$-shift)
  56
  57## Power functions
  58
  59**Strictly increasing:**  $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$)
  60
  61### Odd and even functions
  62Even when $f(x) = -f(x)$  
  63Odd when $-f(x) = f(-x)$
  64
  65Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
  66Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
  67
  68### $x^n$ where $n \in \mathbb{Z}^+$
  69
  70\begin{tabularx}{\textwidth}{|c|c|}
  71  \(n\) is even & \(n\) is odd\\
  72  {\includegraphics[height=1cm]{graphics/parabola.png}} & {\includegraphics[height=1cm]{graphics/cubic.png}}
  73\end{tabularx}
  74
  75### $x^n$ where $n \in \mathbb{Z}^-$
  76
  77\begin{tabularx}{\textwidth}{|c|c|}
  78  \(n\) is even & \(n\) is odd\\
  79  {\includegraphics[height=1cm]{graphics/truncus.png}} & {\includegraphics[height=1cm]{graphics/hyperbola.png}}
  80\end{tabularx}
  81
  82### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
  83
  84\begin{tabularx}{\textwidth}{|c|c|}
  85  \(n\) is even & \(n\) is odd\\
  86  {\includegraphics[height=1cm]{graphics/square-root-graph.png}} & {\includegraphics[height=1cm]{graphics/cube-root-graph.png}}
  87\end{tabularx}
  88
  89### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
  90
  91Mostly only on CAS.
  92
  93We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n.  
  94Domain is:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
  95
  96If $n$ is odd, it is an odd function.
  97
  98### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
  99
 100$$x^{p \over q} = \sqrt[q]{x^p}$$
 101
 102- if $p > q$, the shape of $x^p$ is dominant
 103- if $p < q$, the shape of $x^{1 \over q}$ is dominant
 104- points $(0, 0)$ and $(1, 1)$ will always lie on graph
 105- Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 106
 107
 108## Combinations of functions (piecewise/hybrid)
 109
 110$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 111
 112Open circle - point included  
 113Closed circle - point not included  
 114
 115### Sum, difference, product of functions
 116\begin{tabularx}{\columnwidth}{X|X}
 117  sum & $f+g$ & domain $= \text{dom}(f) \cap \text{dom}(g)$ \\
 118  difference & $f-g$ or $g-f$ & domain $=\text{dom}(f) \cap \text{dom}(g)$ \\
 119  product & $f \times g$ & domain $=\text{dom}(f) \cap \text{dom}(g)$
 120\end{tabularx}
 121  
 122Addition of linear piecewise graphs - add $y$-values at key points
 123
 124Product functions:
 125
 126- product will equal 0 if one of the functions is equal to 0
 127- turning point on one function does not equate to turning point on product
 128
 129## Matrix transformations
 130
 131Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
 132
 133## Composite functions
 134
 135$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
 136
 137