1# Transformation 2 3**Order of operations:** DRT - Dilations, Reflections, Translations 4 5## Transforming $x^n$ to $a(x-h)^n+K$ 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- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis 12 13## Translations 14 15For $y = f(x)$, these processes are equivalent: 16 17- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$ 18- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$ 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## Transforming $f(x)$ to $y=Af[n(x+c)]+b$# 33 34Applies to exponential, log, trig, power, polynomial functions. 35Functions must be written in form $y=Af[n(x+c)] + b$ 36 37$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis) 38$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis) 39$c$ - translation from $y$-axis ($x$-shift) 40$b$ - translation from $x$-axis ($y$-shift) 41 42## Power functions 43 44**Strictly increasing:** $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$) 45 46### Odd and even functions 47Even when $f(x) = -f(x)$ 48Odd when $-f(x) = f(-x)$ 49 50Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$ 51Function $x^{\pm {p \over q}}$ is odd if $q$ is odd 52 53### $x^n$ where $n \in \mathbb{Z}^+$ 54 55| $n$ is even: | $n$ is odd: | 56| ------------ | ----------- | 57|{#id .class width=50%} | {#id .class width=50%} | 58 59### $x^n$ where $n \in \mathbb{Z}^-$ 60 61| $n$ is even: | $n$ is odd: | 62| ------------ | ----------- | 63|{#id .class width=50%} | {#id .class width=50%} | 64 65### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$ 66 67| $n$ is even: | $n$ is odd: | 68| ------------ | ----------- | 69|{#id .class width=50%} | {#id .class width=50%} | 70 71 72### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$ 73 74Mostly only on CAS. 75 76We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n. 77Domain is: $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$ 78 79If $n$ is odd, it is an odd function. 80 81### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$ 82 83$$x^{p \over q} = \sqrt[q]{x^p}$$ 84 85- if $p \gt q$, the shape of $x^p$ is dominant 86- if $p \lt q$, the shape of $x^{1 \over q}$ is dominant 87- points $(0, 0)$ and $(1, 1)$ will always lie on graph 88- Domain is: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$ 89 90 91## Combinations of functions (piecewise/hybrid) 92 93$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0\\2, \hspace{3.4em} 0 < x < 2\\ x, \hspace{3.4em} x \ge 2 \end{cases}$$ 94 95Open circle - point included 96Closed circle - point not included 97 98### Sum, difference, product of functions 99| | | | 100|---|-----|-----| 101|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$| 102|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$| 103|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$| 104 105Addition of linear piecewise graphs - add $y$-values at key points 106 107Product functions: 108- product will equal 0 if one of the functions is equal to 0 109- turning point on one function does not equate to turning point on product 110 111## Matrix transformations 112 113Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$. 114 115## Composite functions 116 117$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$ 118 119