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 $f(-x)=-f(x) \implies f(x)=-f(x)$