methods / transformations.mdon commit [methods] minor methods updates from assignment 4 (091772f)
   1---
   2geometry: margin=2cm
   3columns: 2
   4author: Andrew Lorimer
   5---
   6
   7# Transformation
   8
   9**Order of operations:** DRT - Dilations, Reflections, Translations
  10
  11## Transforming $x^n$ to $a(x-h)^n+K$
  12
  13- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
  14- if $a<0$, graph is reflected over $x$-axis
  15- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
  16- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
  17- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
  18- when $0 < |a| < 1$, graph becomes closer to axis
  19
  20## Dilations
  21
  22For the graph of $y = f(x)$, there are two pairs of equivalent processes:
  23
  241. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
  25   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
  26
  272. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
  28   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
  29
  30For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
  31
  32## Reflections
  33
  34- Reflection **in** axis = reflection **over** axis = reflection **across** axis
  35- Translations do not change
  36
  37## Translations
  38
  39For $y = f(x)$, these processes are equivalent:
  40
  41- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
  42- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
  43
  44## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
  45
  46Applies to exponential, log, trig, power, polynomial functions.  
  47Functions must be written in form $y=Af[n(x+c)] + b$
  48
  49$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)  
  50$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)  
  51$c$ - translation from $y$-axis ($x$-shift)  
  52$b$ - translation from $x$-axis ($y$-shift)
  53
  54## Power functions
  55
  56**Strictly increasing:**  $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$)
  57
  58### Odd and even functions
  59Even when $f(x) = -f(x)$  
  60Odd when $-f(x) = f(-x)$
  61
  62Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
  63Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
  64
  65### $x^n$ where $n \in \mathbb{Z}^+$
  66
  67| $n$ is even: | $n$ is odd: |
  68| ------------ | ----------- |
  69|![](graphics/parabola.png){#id .class width=20%} | ![](graphics/cubic.png){#id .class width=20%} |
  70
  71### $x^n$ where $n \in \mathbb{Z}^-$
  72
  73| $n$ is even: | $n$ is odd: |
  74| ------------ | ----------- |
  75|![](graphics/truncus.png){#id .class width=20%} | ![](graphics/hyperbola.png){#id .class width=20%} |
  76
  77### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
  78
  79| $n$ is even: | $n$ is odd: |
  80| ------------ | ----------- |
  81|![](graphics/square-root-graph.png){#id .class width=20%} | ![](graphics/cube-root-graph.png){#id .class width=20%} |
  82
  83
  84### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
  85
  86Mostly only on CAS.
  87
  88We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n.  
  89Domain is:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
  90
  91If $n$ is odd, it is an odd function.
  92
  93### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
  94
  95$$x^{p \over q} = \sqrt[q]{x^p}$$
  96
  97- if $p > q$, the shape of $x^p$ is dominant
  98- if $p < q$, the shape of $x^{1 \over q}$ is dominant
  99- points $(0, 0)$ and $(1, 1)$ will always lie on graph
 100- Domain is:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
 101
 102
 103## Combinations of functions (piecewise/hybrid)
 104
 105$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
 106
 107Open circle - point included  
 108Closed circle - point not included  
 109
 110### Sum, difference, product of functions
 111| | | |
 112|---|-----|-----|
 113|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$|
 114|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
 115|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
 116
 117Addition of linear piecewise graphs - add $y$-values at key points
 118
 119Product functions:
 120
 121- product will equal 0 if one of the functions is equal to 0
 122- turning point on one function does not equate to turning point on product
 123
 124## Matrix transformations
 125
 126Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
 127
 128## Composite functions
 129
 130$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
 131
 132