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