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# Transformations

**Order of operations:** DRT

\begin{center}dilations --- reflections --- translations\end{center}

## Transforming $x^n$ to $a(x-h)^n+K$

- dilation factor of $|a|$ units parallel to $y$-axis or from $x$-axis
- if $a<0$, graph is reflected over $x$-axis
- translation of $k$ units parallel to $y$-axis or from $x$-axis
- 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

## Transforming $f(x)$ to $y=Af[n(x+c)]+b$#

Applies to exponential, log, trig, $e^x$, polynomials.  
Functions must be written in form $y=Af[n(x+c)]+b$

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

## Dilations

Two pairs of equivalent processes for $y=f(x)$:

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.

## Matrix transformations

Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$.

## 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 $y-k = f(x-h)$

## 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


\newcolumntype{C}{>{\centering\arraybackslash} m{3cm} }
\begin{center}
\begin{tabular}{m{1.2cm}|C|C}
  & $n$ is even & $n$ is odd \\
  \hline
  \parbox[c]{1.2cm}{$x^n,\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/parabola.png}} & {\includegraphics[height=3cm]{graphics/cubic.png}}\\
  \parbox[c]{1.2cm}{$x^n$,\\ $n \in \mathbb{Z}^-$} & {\includegraphics[height=3cm]{graphics/truncus.png}} & {\includegraphics[height=3cm]{graphics/hyperbola.png}}\\
  \parbox[c]{1.2cm}{$x^{1 \over n},\\ n \in \mathbb{Z}^+$} & {\includegraphics[height=3cm]{graphics/square-root-graph.png}} & {\includegraphics[height=3cm]{graphics/cube-root-graph.png}}\\
\end{tabular}
\end{center}

### $x^{-1 \over n}$ 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.

\columnbreak

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

## Piecewise functions

$$\text{e.g.} \quad f(x) = \begin{cases} x^{1 / 3}, \hspace{2em} x \le 0 \\ 2, \hspace{3.4em} 0 < x < 2 \\ x, \hspace{3.4em} x \ge 2 \end{cases}$$

**Open circle:** point included  
**Closed circle:** point not included

## Operations on functions

For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$

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

Product functions:

- product will equal 0 if $f=0$ or $g=0$
- $f^\prime(x)=0 \veebar g^\prime(x)=0 \not\Rightarrow (f \times g)^\prime(x)=0$

## Composite functions

$(f \circ g)(x)$ is defined iff $\operatorname{ran}(g) \subseteq \operatorname{dom}(f)$