From: Andrew Lorimer <andrew@lorimer.id.au>
Date: Fri, 15 Feb 2019 05:07:25 +0000 (+1100)
Subject: finish methods notes on x^n functions
X-Git-Tag: yr12~253
X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/e0cc0b5fe573ce7af6b3ca4d17b449e6dcd9c5cf

finish methods notes on x^n functions
---

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diff --git a/methods/transformations.md b/methods/transformations.md
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--- a/methods/transformations.md
+++ b/methods/transformations.md
@@ -2,18 +2,19 @@
 
 **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
 
 ## 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
@@ -28,7 +29,7 @@ For the graph of $y = f(x)$, there are two pairs of equivalent processes:
 
 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 $y=Af[n(x+c)] + b$
@@ -40,23 +41,52 @@ $b$ - translation from $x$-axis ($y$-shift)
 
 ## 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 $x=0$)
 
-#### $n$ is odd and $n>1$:  
-$f(-x)=-f(x)$
+### Odd and even functions
+Even when $f(x) = -f(x)$  
+Odd when $-f(x) = f(-x)$
 
-#### $n$ is even and $n>1$:
-$f(-x)=f(x)$
+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=50%} | ![](graphics/cubic.png){#id .class width=50%} |
+
+### $x^n$ where $n \in \mathbb{Z}^-$
+
+| $n$ is even: | $n$ is odd: |
+| ------------ | ----------- |
+|![](graphics/truncus.png){#id .class width=50%} | ![](graphics/hyperbola.png){#id .class width=50%} |
+
+### $x^{1 \over n}$ where $n \in \mathbb{Z}^+$
+
+| $n$ is even: | $n$ is odd: |
+| ------------ | ----------- |
+|![](graphics/square-root-graph.png){#id .class width=50%} | ![](graphics/cube-root-graph.png){#id .class width=50%} |
 
-### Function $f(x)=x^{-1 \over n}$ where $n \in \mathbb{Z}^+$
+
+### $x^{-1 \over n}$ 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 $p \gt q$, the shape of $x^p$ is dominant
+- if $p \lt q$, 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)