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# Transformation

**Order of operations:** DRT - Dilations, Reflections, Translations

## 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

## Dilations

For the graph of $y = f(x)$, there are two pairs of equivalent processes:

1. - Dilating from $x$-axis: $(x, y) \rightarrow (x, by)$
   - Replacing $y$ with $y \over b$ to obtain $y = b f(x)$

2. - Dilating from $y$-axis: $(x, y) \rightarrow (ax, y)$
   - Replacing $x$ with $x \over a$ to obtain $y = f({x \over a})$

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.  
Functions must be written in form $y=Af[n(x+c)] + b$

$A$ - dilation by factor $A$ from $x$-axis (if $A<0$, reflection across $y$-axis)  
$n$ - dilation by factor $1 \over n$ from $y$-axis (if $n<0$, reflection across $x$-axis)  
$c$ - translation from $y$-axis ($x$-shift)  
$b$ - translation from $x$-axis ($y$-shift)

## Power functions

**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: |
| ------------ | ----------- |
|![](graphics/parabola.png){#id .class width=20%} | ![](graphics/cubic.png){#id .class width=20%} |

### $x^n$ where $n \in \mathbb{Z}^-$

| $n$ is even: | $n$ is odd: |
| ------------ | ----------- |
|![](graphics/truncus.png){#id .class width=20%} | ![](graphics/hyperbola.png){#id .class width=20%} |

### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$

| $n$ is even: | $n$ is odd: |
| ------------ | ----------- |
|![](graphics/square-root-graph.png){#id .class width=20%} | ![](graphics/cube-root-graph.png){#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:  $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n\text{ is odd} \\ \mathbb{R}^+ \hspace{2.6em}\text{if }n\text{ is even}\end{cases}$

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:  $\begin{cases} \mathbb{R} \hspace{4em}\text{ if }q\text{ is odd} \\ \mathbb{R}^+ \cup \{0\} \hspace{1em}\text{if }q\text{ is even}\end{cases}$


## Combinations of functions (piecewise/hybrid)

$$\text{e.g.}\quad f(x)=\begin{cases} ^3 \sqrt{x}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$

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 $=\text{dom}(f) \cap \text{dom}(g)$|

Addition of linear piecewise graphs - add $y$-values at key points

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

## 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 $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$