From e0cc0b5fe573ce7af6b3ca4d17b449e6dcd9c5cf Mon Sep 17 00:00:00 2001 From: Andrew Lorimer Date: Fri, 15 Feb 2019 16:07:25 +1100 Subject: [PATCH] finish methods notes on x^n functions --- methods/graphics/cube-root-graph.png | Bin 0 -> 42940 bytes methods/graphics/cubic.png | Bin 0 -> 48640 bytes methods/graphics/hyperbola.png | Bin 0 -> 46693 bytes methods/graphics/parabola.png | Bin 0 -> 51261 bytes methods/graphics/square-root-graph.png | Bin 0 -> 36624 bytes methods/graphics/truncus.png | Bin 0 -> 45611 bytes methods/graphics/xtotwothirds.png | Bin 0 -> 43277 bytes methods/transformations.md | 56 +++++++++++++++++++------ 8 files changed, 43 insertions(+), 13 deletions(-) create mode 100644 methods/graphics/cube-root-graph.png create mode 100644 methods/graphics/cubic.png create mode 100644 methods/graphics/hyperbola.png create mode 100644 methods/graphics/parabola.png create mode 100644 methods/graphics/square-root-graph.png create mode 100644 methods/graphics/truncus.png create 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$f(x)=a(x-h)^n+K$## +## 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) -- 2.43.2