+---
+geometry: margin=2cm
+author: Andrew Lorimer
+---
+
# Transformation
**Order of operations:** DRT - Dilations, Reflections, Translations
-## $f(x) = x^n$ to $f(x)=a(x-h)^n+K$##
+## Transforming $x^n$ to $a(x-h)^n+K$
- $|a|$ is the dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
- if $a<0$, graph is reflected over $x$-axis
- $k$ - translation of $k$ units parallel to $y$-axis or from $x$-axis
- $h$ - translation of $h$ units parallel to $x$-axis or from $y$-axis
+- for $(ax)^n$, dilation factor is $1 \over a$ parallel to $x$-axis or from $y$-axis
+- when $0 < |a| < 1$, graph becomes closer to axis
## Translations
For $y = f(x)$, these processes are equivalent:
-- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f$(x)$
+- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
## Dilations
For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (symmetrical). If $y={a \over x}$, graph is contracted rather than dilated.
-## Transformations from $f(x)$ to $y=Af[n(x+c)]+b$#
+## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
Applies to exponential, log, trig, power, polynomial functions.
Functions must be written in form $y=Af[n(x+c)] + b$
## Power functions
-**Strictly increasing** on an interval where $x_2 > x_1 \implies f(x_2) > f(x_2)$ (including $x=0$)
+**Strictly increasing:** $f(x_2) > f(x_1)$ where $x_2 > x_1$ (including $x=0$)
+
+### Odd and even functions
+Even when $f(x) = -f(x)$
+Odd when $-f(x) = f(-x)$
+
+Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$
+Function $x^{\pm {p \over q}}$ is odd if $q$ is odd
+
+### $x^n$ where $n \in \mathbb{Z}^+$
+
+| $n$ is even: | $n$ is odd: |
+| ------------ | ----------- |
+|{#id .class width=20%} | {#id .class width=20%} |
-#### $n$ is odd and $n>1$:
-$f(-x)=-f(x)$
+### $x^n$ where $n \in \mathbb{Z}^-$
-#### $n$ is even and $n>1$:
-$f(-x)=f(x)$
+| $n$ is even: | $n$ is odd: |
+| ------------ | ----------- |
+|{#id .class width=20%} | {#id .class width=20%} |
-### Function $f(x)=x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
+### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
+
+| $n$ is even: | $n$ is odd: |
+| ------------ | ----------- |
+|{#id .class width=20%} | {#id .class width=20%} |
+
+
+### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
Mostly only on CAS.
-We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n. Domain is: $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
+We can write $x^{-1 \over n} = {1 \over {x^{1 \over n}}} = {1 \over ^n \sqrt{x}}$n.
+Domain is: $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$
+
+If $n$ is odd, it is an odd function.
+
+### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$
+
+$$x^{p \over q} = \sqrt[q]{x^p}$$
+
+- if $p > q$, the shape of $x^p$ is dominant
+- if $p < q$, the shape of $x^{1 \over q}$ is dominant
+- points $(0, 0)$ and $(1, 1)$ will always lie on graph
+- Domain is: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
-**Odd and even functions:**
-Function is even if it can be reflected across $y$-axis $\implies f(x)=f(-x)$
-If $n$ is odd, then $f$ is an odd function since $f(-x)=-f(x) \implies f(x)=-f(x)$
## Combinations of functions (piecewise/hybrid)
Addition of linear piecewise graphs - add $y$-values at key points
-Product functions:
+Product functions:
+
- product will equal 0 if one of the functions is equal to 0
-- turning point on one function does not equate to turning point on product
\ No newline at end of file
+- turning point on one function does not equate to turning point on product
+
+## Matrix transformations
+
+Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.
+
+## Composite functions
+
+$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$
+
+