methods / transformations.mdon commit piecewise functions (5247004)
   1# Transformation
   2
   3**Order of operations:** DRT - Dilations, Reflections, Translations
   4
   5## $f(x) = x^n$ to $f(x)=a(x-h)^n+K$##
   6
   7- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
   8- if $a<0$, graph is reflected over $x$-axis
   9- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
  10- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
  11
  12## Translations
  13
  14For $y = f(x)$, these processes are equivalent:
  15
  16- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f$(x)$
  17- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
  18
  19## Dilations
  20
  21For the graph of $y = f(x)$, there are two pairs of equivalent processes:
  22
  231. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
  24   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$
  25
  262. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
  27   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$
  28
  29For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
  30
  31## Transformations from $f(x)$ to $y=Af[n(x+c)]+b$#
  32
  33Applies to exponential, log, trig, power, polynomial functions.  
  34Functions must be written in form $y=Af[n(x+c)] + b$
  35
  36$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)  
  37$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)  
  38$c$ - translation from $y$-axis ($x$-shift)  
  39$b$ - translation from $x$-axis ($y$-shift)
  40
  41## Power functions
  42
  43**Strictly increasing** on an interval where $x_2 > x_1 \implies f(x_2) > f(x_2)$ (including $x=0$)
  44
  45#### $n$ is odd and $n>1$:  
  46$f(-x)=-f(x)$
  47
  48#### $n$ is even and $n>1$:
  49$f(-x)=f(x)$
  50
  51### Function $f(x)=x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
  52
  53Mostly only on CAS.
  54
  55We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n. Domain is:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
  56
  57**Odd and even functions:**  
  58Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$  
  59If $n$ is odd, then $f$ is an odd function since $f(-x)=-f(x) \implies f(x)=-f(x)$
  60
  61## Combinations of functions (piecewise/hybrid)
  62
  63$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
  64
  65Open circle - point included  
  66Closed circle - point not included  
  67
  68### Sum, difference, product of functions
  69| | | |
  70|---|-----|-----|
  71|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$|
  72|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
  73|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
  74
  75Addition of linear piecewise graphs - add $y$-values at key points
  76
  77Product functions:  
  78- product will equal 0 if one of the functions is equal to 0
  79- turning point on one function does not equate to turning point on product