+---
+geometry: margin=2cm
+columns: 2
+author: Andrew Lorimer
+header-includes:
+- \usepackage{graphicx}
+- \usepackage{tabularx}
+---
+
# Transformation
**Order of operations:** DRT - Dilations, Reflections, Translations
- $k$ - 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
-
-## Translations
-
-For $y = f(x)$, these processes are equivalent:
-
-- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
-- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+- when $0 < |a| < 1$, graph becomes closer to axis
## 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.
+## Reflections
+
+- Reflection **in** axis = reflection **over** axis = reflection **across** axis
+- Translations do not change
+
+## Translations
+
+For $y = f(x)$, these processes are equivalent:
+
+- applying the translation $(x, y) \rightarrow (x + h, y + k)$ to the graph of $y = f(x)$
+- replacing $x$ with $x − h$ and $y$ with $y − k$ to obtain $y − k = f (x − h)$
+
## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#
Applies to exponential, log, trig, power, polynomial functions.
### $x^n$ where $n \in \mathbb{Z}^+$
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/parabola.png}} & {\includegraphics[height=1cm]{graphics/cubic.png}}
+\end{tabularx}
### $x^n$ where $n \in \mathbb{Z}^-$
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/truncus.png}} & {\includegraphics[height=1cm]{graphics/hyperbola.png}}
+\end{tabularx}
### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
-| $n$ is even: | $n$ is odd: |
-| ------------ | ----------- |
-|{#id .class width=50%} | {#id .class width=50%} |
-
+\begin{tabularx}{\textwidth}{|c|c|}
+ \(n\) is even & \(n\) is odd\\
+ {\includegraphics[height=1cm]{graphics/square-root-graph.png}} & {\includegraphics[height=1cm]{graphics/cube-root-graph.png}}
+\end{tabularx}
### $x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
$$x^{p \over q} = \sqrt[q]{x^p}$$
-- if $p \gt q$, the shape of $x^p$ is dominant
-- if $p \lt q$, the shape of $x^{1 \over q}$ is dominant
+- 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: $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$
## Combinations of functions (piecewise/hybrid)
-$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
+$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$
Open circle - point included
Closed circle - point not included
### Sum, difference, product of functions
-| | | |
-|---|-----|-----|
-|sum|$f+g$|domain $= \text{dom}(f) \cap \text{dom}(g)$|
-|difference|$f-g$ or $g-f$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
-|product|$f \times g$|domain $=\text{dom}(f) \cap \text{dom}(g)$|
-
+\begin{tabularx}{\columnwidth}{X|X}
+ sum & $f+g$ & domain $= \text{dom}(f) \cap \text{dom}(g)$ \\
+ difference & $f-g$ or $g-f$ & domain $=\text{dom}(f) \cap \text{dom}(g)$ \\
+ product & $f \times g$ & domain $=\text{dom}(f) \cap \text{dom}(g)$
+\end{tabularx}
+
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