From: Andrew Lorimer Date: Mon, 13 May 2019 11:17:47 +0000 (+1000) Subject: [methods] fix formatting of notes X-Git-Tag: yr12~139 X-Git-Url: https://git.lorimer.id.au/notes.git/diff_plain/83f70da25a68a2aa7461955d9f60ce8249053d7c?ds=inline [methods] fix formatting of notes --- diff --git a/methods/calculus-ref.md b/methods/calculus-ref.md index 215845d..e12b37e 100644 --- a/methods/calculus-ref.md +++ b/methods/calculus-ref.md @@ -1,25 +1,29 @@ --- -geometry: margin=1cm +geometry: a4paper, margin=2cm columns: 2 -graphics: yes -tables: yes author: Andrew Lorimer header-includes: +- \usepackage{fancyhdr} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} +- \usepackage{graphicx} - \usepackage{tabularx} +- \usepackage[dvipsnames, table]{xcolor} --- - +\linespread{3} \pagenumbering{gobble} \renewcommand{\arraystretch}{1.4} +\definecolor{cas}{HTML}{e6f0fe} - -# Methods - Calculus +# Calculus ## Average rate of change $$m \operatorname{of} x \in [a,b] = {{f(b)-f(a)}\over {b - a}} = {dy \over dx}$$ -On CAS: Action $\rightarrow$ Calculation $\rightarrow$ Diff $\rightarrow$ ($f(x)$ | $y$) $=\dots$ +\colorbox{cas}{On CAS:} Action $\rightarrow$ Calculation $\rightarrow$ `diff` ## Instantaneous rate of change @@ -51,18 +55,22 @@ Not differentiable at: **Normal line** - $\perp$ tangent ($m_{{tan}} \cdot m_{\operatorname{norm}} = -1$) **Secant** $={{f(x+h)-f(x)} \over h}$ -## Strictly increasing +## Strictly increasing/decreasing -- **strictly increasing** where $f(x_2) > f(x_1)$ and $x_2 > x_1$ -- **strictly decreasing** where $f(x_2) < f(x_1)$ and $x_2 > x_1$ -- If $f^\prime (x) > 0$ for all $x$ in interval, then $f$ is **strictly increasing** -- If $f^\prime(x) < 0$ for all $x$ in interval, then $f$ is **strictly decreasing** +For $x_2$ and $x_1$ where $x_2 > x_1$: + +- **strictly increasing** where $f(x_2) > f(x_1)$ +or $f^\prime(x)>0$ +- **strictly decreasing** where $f(x_2) < f(x_1)$ +or $f^\prime(x)<0$ - Endpoints are included, even where gradient $=0$ +\columnbreak + ### Solving on CAS -**In main**: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line) -**In graph**: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line. +\colorbox{cas}{\textbf{In main}}: type function. Interactive $\rightarrow$ Calculation $\rightarrow$ Line $\rightarrow$ (Normal | Tan line) +\colorbox{cas}{\textbf{In graph}}: define function. Analysis $\rightarrow$ Sketch $\rightarrow$ (Normal | Tan line). Type $x$ value to solve for a point. Return to show equation for line. ## Stationary points @@ -87,11 +95,14 @@ Find derivative, solve for ${dy \over dx} = 0$ ## Function derivatives -\begin{tabularx}{\columnwidth}{rl} +\definecolor{shade1}{HTML}{ffffff} +\definecolor{shade2}{HTML}{F0F9E4} +\rowcolors{1}{shade1}{shade2} +\begin{tabularx}{\columnwidth}{rX} \hline \(f(x)\) & \(f^\prime(x)\) \\ \hline - \(kx^n\) & \(knx^{n-1}\)\tabularnewline + \hspace{6em} \(kx^n\) & \(knx^{n-1}\)\tabularnewline \(g(x) \pm h(x)\) & \(g^\prime (x) \pm h^\prime (x)\)\tabularnewline \(c\) & \(0\)\tabularnewline \({u \over v}\) & diff --git a/methods/calculus-ref.pdf b/methods/calculus-ref.pdf index 06fb928..fa940f3 100644 Binary files a/methods/calculus-ref.pdf and b/methods/calculus-ref.pdf differ diff --git a/methods/circ-functions.md b/methods/circ-functions.md index e639e70..27ce6d5 100644 --- a/methods/circ-functions.md +++ b/methods/circ-functions.md @@ -1,21 +1,27 @@ --- -geometry: margin=2cm +geometry: a4paper, margin=2cm columns: 2 -graphics: yes +author: Andrew Lorimer +header-includes: +- \usepackage{setspace} +- \usepackage{fancyhdr} +- \usepackage{graphicx} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} --- -# Circular functions - +\setstretch{1.2} +\pagenumbering{gobble} - +# Circular functions ## Exact values \includegraphics[width=0.2\textwidth]{./graphics/exact-values-1.png} \includegraphics[width=0.2\textwidth]{./graphics/exact-values-2.png} - - +$$1 \thinspace \operatorname{rad}={{180 \operatorname{deg}}\over \pi}$$ ## $\sin$ and $\cos$ graphs diff --git a/methods/circ-functions.pdf b/methods/circ-functions.pdf index 84f3121..990d0dc 100644 Binary files a/methods/circ-functions.pdf and b/methods/circ-functions.pdf differ diff --git a/methods/inverse-functions.md b/methods/inverse-functions.md index 6710e9c..8b4de94 100644 --- a/methods/inverse-functions.md +++ b/methods/inverse-functions.md @@ -1,3 +1,17 @@ +--- +geometry: margin=2cm +author: Andrew Lorimer +header-includes: +- \usepackage{setspace} +- \usepackage{fancyhdr} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} +--- + +\setstretch{1.3} +\pagenumbering{gobble} + # Inverse functions ## Functions @@ -7,24 +21,24 @@ ## One to one functions -- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$ -- i.e. unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is) +- $f(x)$ is *one to one* if $f(a) \ne f(b)$ if $a, b \in \operatorname{dom}(f)$ and $a \ne b$ +$\implies$ unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is) - horizontal line test - if not one to one, it is many to one -## Inverse functions $f^{-1}$ +## Deriving $f^{-1}$ - if $f(g(x)) = x$, then $g$ is the inverse of $f$ - reflection across $y-x$ - $\operatorname{ran} \> f = \operatorname{dom} \> f^{-1}, \quad \operatorname{dom} \> f = \operatorname{ran} \> f^{-1}$ -- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test) -- - $\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one +- inverse $\ne$ inverse *function* (i.e. inverse must pass vertical line test) +$\implies f^{-1}(x)$ exists $\iff f(x)$ is one to one - $f^{-1}(x)=f(x)$ intersections may lie on line $y=x$ -Requirements for showing working for $f^{-1}$: +### Requirements for showing working for $f^{-1}$ -- start with *"let $y=f(x)$"* -- must state *"take inverse"* for line where $y$ and $x$ are swapped -- do all working in terms of $y=\dots$ -- for square root, state $\pm$ solutions then show restricted -- for inverse *function*, state in function notation +1. start with *"let $y=f(x)$"* +2. must state *"take inverse"* for line where $y$ and $x$ are swapped +3. do all working in terms of $y=\dots$ +4. for square root, state $\pm$ solutions then show restricted +5. for inverse *function*, state in function notation diff --git a/methods/inverse-functions.pdf b/methods/inverse-functions.pdf index 5e5f1f8..a5e91c4 100644 Binary files a/methods/inverse-functions.pdf and b/methods/inverse-functions.pdf differ diff --git a/methods/polynomials-ref.pdf b/methods/polynomials-ref.pdf index 98b6422..4840812 100644 Binary files a/methods/polynomials-ref.pdf and b/methods/polynomials-ref.pdf differ diff --git a/methods/polynomials.md b/methods/polynomials.md index c826306..6a4cf98 100644 --- a/methods/polynomials.md +++ b/methods/polynomials.md @@ -1,10 +1,20 @@ --- -geometry: margin=1.5cm +geometry: a4paper, margin=2cm columns: 2 +author: Andrew Lorimer header-includes: +- \usepackage{setspace} +- \usepackage{fancyhdr} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} +- \usepackage{graphicx} - \usepackage{tabularx} +- \usepackage[dvipsnames]{xcolor} --- +\setstretch{1.3} +\definecolor{cas}{HTML}{e6f0fe} \pagenumbering{gobble} \renewcommand{\arraystretch}{1.4} @@ -13,12 +23,15 @@ header-includes: ## Quadratics \newcolumntype{R}{>{\raggedleft\arraybackslash}X} -\begin{tabularx}{\columnwidth}{|R|l|} - Quadratics & $x^2 + bx + c = (x+m)(x+n)$ \\ - & where $mn=c, \> m+n=b$ \\ +\begin{tabularx}{\columnwidth}{Rl} + General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\ + \hline Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\ - Perfect squares & $a^2 \pm 2ab + b^2 = (a \pm b^2)$ \\ + \hline + Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\ + \hline Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\ + \hline Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\ \end{tabularx} @@ -28,6 +41,13 @@ header-includes: **Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ **Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$ +$$y=a(bx-h)^3 + c$$ + +- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$) +- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$ +- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$ +- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$ + ## Linear and quadratic graphs ### Forms of linear equations @@ -40,17 +60,7 @@ $y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts Parallel lines: $m_1 = m_2$ Perpendicular lines: $m_1 \times m_2 = -1$ -Distance: $\vec{AB} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ - - -## Cubic graphs - -$$y=a(bx-h)^3 + c$$ - -- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$) -- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$ -- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$ -- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$ +Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ ## Quartic graphs @@ -62,17 +72,13 @@ $y=ax^2(x-b)(x-c)$ $y=a(x-b)^2(x-c)^2$ $y=a(x-b)(x-c)^3$ -## Literal equations - -Equations with multiple pronumerals. Solutions are expressed in terms of pronumerals (parameters) - ## Simultaneous equations (linear) - **Unique solution** - lines intersect at point - **Infinitely many solutions** - lines are equal - **No solution** - lines are parallel -### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases}$ for one, infinite and no solutions +### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>$ for $\{0,1,\infty\}$ solutions where all coefficients are known except for one, and $a, b$ are known @@ -87,10 +93,10 @@ where all coefficients are known except for one, and $a, b$ are known - *--- for infinite/no solutions: ---* 5. Substitute variable into both original equations 6. Rearrange equations so that LHS of each is the same -7. If $\text{RHS}(1) = \text{RHS}(2)$, lines are coincident (infinite solutions) - If $\text{RHS}(1) \ne \text{RHS}(2)$, lines are parallel (no solutions) +7. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns) + $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns) -Or use Matrix -> `det` on CAS. +\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det` ### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ a_2 x + b_2 y + c_2 z = d_2 \\ diff --git a/methods/stuff.md b/methods/stuff.md index 2055b63..d48dadb 100644 --- a/methods/stuff.md +++ b/methods/stuff.md @@ -1,79 +1,69 @@ --- -geometry: margin=1.5cm - -graphics: yes -tables: yes +geometry: a4paper, margin=2cm +columns: 2 author: Andrew Lorimer -classoption: twocolumn -header-includes: \pagenumbering{gobble} +header-includes: +- \usepackage{fancyhdr} +- \usepackage{setspace} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} +- \usepackage{graphicx} +- \usepackage{tabularx} +- \usepackage[dvipsnames]{xcolor} --- -# Exponential and Index Functions +\pagenumbering{gobble} +\setstretch{1.5} +\definecolor{cas}{HTML}{e6f0fe} + +# Exponentials & Logarithms ## Index laws -\begin{equation}\begin{split} +\begin{equation*}\begin{split} a^m \times a^n & = a^{m+n} \\ - a^m \div a^n & = a^{m-n}4 \\ + a^m \div a^n & = a^{m-n} \\ (a^m)^n & = a^{_mn} \\ (ab)^m & = a^m b^m \\ - {({a \over b})}^m & = {a^m \over b^m} -\end{split}\end{equation} - -## Fractional indices - -$$^n\sqrt{x}=x^{1/n}$$ - -## Logarithms - -$$\log_b (x) = n \quad \operatorname{where} \hspace{0.5em} b^n=x$$ - -## Using logs to solve index eq's - -Used for equations without common base exponent - -Or change base: -$$\log_b c = {{\log_a c} \over {\log_a b}}$$ - -If $a<1, \quad \log_{b} a < 0$ (flip inequality operator) - -## Exponential functions - -$e^x$ - natural exponential function - -$$\lim_{h \rightarrow 0} {{e^h-1} \over h}=1$$ + {({a \over b})}^m & = {a^m \over b^m} \\ + ^n\sqrt{x} &=x^{1/n} +\end{split}\end{equation*} ## Logarithm laws -\begin{equation}\begin{split} +\begin{equation*}\begin{split} \log_a(mn) & = \log_am + \log_an \\ \log_a({m \over n}) & = \log_am - \log_a \\ \log_a(m^p) & = p\log_am \\ \log_a(m^{-1}) & = -\log_am \\ - \log_a1 = 0 & \text{ and } \log_aa = 1 -\end{split}\end{equation} - + \log_a1 = 0 & \text{ and } \log_aa = 1 \\ + \log_b c &= {{\log_a c} \over {\log_a b}} +\end{split}\end{equation*} ## Inverse functions For $f: \mathbb{R} \rightarrow \mathbb{R}, f(x)=a^x$, inverse is: -$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=log_ax$$ +$$f^{-1}: \mathbb{R}^+ \rightarrow \mathbb{R}, f^{-1}=\log_ax$$ + +## Exponentials -## Euler's number +$$e^x \quad \text{natural exponential function}$$ $$e= \lim_{n \rightarrow \infty} (1 + {1 \over n})^n$$ -## Exponential and logarithmic modelling +## Modelling $$A = A_0 e^{kt}$$ -where -$A_0$ is initial value -$t$ is time taken -$k$ is a constant -For continuous growth, $k > 0$ -For continuous decay, $k < 0$ +- $A_0$ is initial value +- $t$ is time taken +- $k$ is a constant +- For continuous growth, $k > 0$ +- For continuous decay, $k < 0$ + +\columnbreak ## Graphing exponential functions @@ -83,7 +73,7 @@ $$f(x)=Aa^{k(x-b)} + c, \quad \vert \> a > 1$$ - **horizontal asymptote** at $y=c$ - **domain** is $\mathbb{R}$ - **range** is $(c, \infty)$ -- dilation of factor $A$ from $x$-axis +- dilation of factor $|A|$ from $x$-axis - dilation of factor $1 \over k$ from $y$-axis ![](graphics/exponential-graphs.png){#id .class width=30%} @@ -100,11 +90,11 @@ where - **range** is $\mathbb{R}$ - **vertical asymptote** at $x=b$ - $y$-intercept exists if $b<0$ -- dilation of factor $A$ from $x$-axis +- dilation of factor $|A|$ from $x$-axis - dilation of factor $1 \over k$ from $y$-axis ![](graphics/log-graphs.png){#id .class width=30%} ## Finding equations -Solve simultaneous equations on CAS: ![](graphics/cas-simultaneous.png){#id .class width=75px} +\colorbox{cas}{On CAS:} ![](graphics/cas-simultaneous.png){#id .class width=75px} diff --git a/methods/stuff.pdf b/methods/stuff.pdf index 2de4e75..e7aff37 100644 Binary files a/methods/stuff.pdf and b/methods/stuff.pdf differ diff --git a/methods/transformations-ref.pdf b/methods/transformations-ref.pdf index bbcf953..949f92f 100644 Binary files a/methods/transformations-ref.pdf and b/methods/transformations-ref.pdf differ diff --git a/methods/transformations.md b/methods/transformations.md index 0378cae..b04c717 100644 --- a/methods/transformations.md +++ b/methods/transformations.md @@ -1,28 +1,48 @@ --- -geometry: margin=2cm +geometry: a4paper, margin=2cm columns: 2 author: Andrew Lorimer header-includes: +- \usepackage{setspace} +- \usepackage{fancyhdr} +- \pagestyle{fancy} +- \fancyhead[LO,LE]{Year 12 Methods} +- \fancyhead[CO,CE]{Andrew Lorimer} - \usepackage{graphicx} - \usepackage{tabularx} --- -# Transformation +\setstretch{1.6} +\pagenumbering{gobble} -**Order of operations:** DRT - Dilations, Reflections, Translations +# Transformations + +**Order of operations:** DRT + +\begin{center}dilations --- reflections --- translations\end{center} ## 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 +- 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 +- 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 -For the graph of $y = f(x)$, there are two pairs of equivalent processes: +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 $y = b f(x)$ @@ -32,6 +52,10 @@ 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. +## 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 @@ -42,17 +66,7 @@ For graph of $y={1 \over x}$, horizontal & vertical dilations are equivalent (sy 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) +- replacing $x$ with $x-h$ and $y$ with $y-k$ to obtain $y-k = f(x-h)$ ## Power functions @@ -65,26 +79,17 @@ 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 $n \in \mathbb{Z}^+$ - -\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 $n \in \mathbb{Z}^-$ - -\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 $n \in \mathbb{Z}^+$ -\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} +\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 $n \in \mathbb{Z}^+$ @@ -95,6 +100,8 @@ Domain is: $\begin{cases} \mathbb{R} \setminus \{0\}\hspace{0.5em} \text{ if }n If $n$ is odd, it is an odd function. +\columnbreak + ### $x^{p \over q}$ where $p, q \in \mathbb{Z}^+$ $$x^{p \over q} = \sqrt[q]{x^p}$$ @@ -104,31 +111,23 @@ $$x^{p \over q} = \sqrt[q]{x^p}$$ - 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) +## 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 +**Open circle:** point included +**Closed circle:** point not included -### Sum, difference, product of functions -\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 +## Operations on functions -Product functions: +For $f \pm g$ and $f \times g$: \quad $\text{dom}^\prime = \operatorname{dom}(f) \cap \operatorname{dom}(g)$ -- 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 +Addition of linear piecewise graphs: add $y$-values at key points -## Matrix transformations +Product functions: -Find new point $(x^\prime, y^\prime)$. Substitute these into original equation to find image with original variables $(x, y)$. +- 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