1--- 2geometry: a4paper, margin=2cm 3columns: 2 4author: Andrew Lorimer 5header-includes: 6- \usepackage{setspace} 7- \usepackage{fancyhdr} 8- \pagestyle{fancy} 9-\fancyhead[LO,LE]{Year 12 Methods} 10-\fancyhead[CO,CE]{Andrew Lorimer} 11- \usepackage{graphicx} 12- \usepackage{tabularx} 13- \usepackage[dvipsnames]{xcolor} 14--- 15 16\setstretch{1.3} 17\definecolor{cas}{HTML}{e6f0fe} 18\pagenumbering{gobble} 19\renewcommand{\arraystretch}{1.4} 20 21# Polynomials 22 23## Quadratics 24 25\newcolumntype{R}{>{\raggedleft\arraybackslash}X} 26\begin{tabularx}{\columnwidth}{Rl} 27 General form& \parbox[t]{5cm}{$x^2 + bx + c = (x+m)(x+n)$\\ where $mn=c, \> m+n=b$} \\ 28 \hline 29 Difference of squares & $a^2 - b^2 = (a - b)(a + b)$ \\ 30 \hline 31 Perfect squares & \parbox[c]{5cm}{$a^2 \pm 2ab + b^2 = (a \pm b^2)$} \\ 32 \hline 33 Completing the square & \parbox[t]{5cm}{$x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ \\ $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$} \\ 34 \hline 35 Quadratic formula & $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ where $\Delta=b^2-4ac$ \\ 36\end{tabularx} 37 38## Cubics 39 40**Difference of cubes:** $a^3 - b^3 = (a-b)(a^2 + ab + b^2)$ 41**Sum of cubes:** $a^3 + b^3 = (a+b)(a^2 - ab + b^2)$ 42**Perfect cubes:** $a^3 \pm 3a^2b + 3ab^2 \pm b^3 = (a \pm b)^3$ 43 44$$y=a(bx-h)^3 + c$$ 45 46- $m=0$ at *stationary point of inflection* (i.e. (${h \over b}, k)$) 47- in form $y=(x-a)^2(x-b)$, local max at $x=a$, local min at $x=b$ 48- in form $y=a(x-b)(x-c)(x-d)$: $x$-intercepts at $b, c, d$ 49- in form $y=a(x-b)^2(x-c)$, touches $x$-axis at $b$, intercept at $c$ 50 51## Linear and quadratic graphs 52 53### Forms of linear equations 54 55$y=mx+c$ where $m$ is gradient and $c$ is $y$-intercept 56${x \over a} + {y \over b}=1$ where $m$ is gradient and $(x_1, y_1)$ lies on the graph 57$y-y_1 = m(x-x_1)$ where $(a,0)$ and $(0,b)$ are $x$- and $y$-intercepts 58 59## Line properties 60 61Parallel lines: $m_1 = m_2$ 62Perpendicular lines: $m_1 \times m_2 = -1$ 63Distance: $|\vec{AB}| = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ 64 65## Quartic graphs 66 67### Forms of quadratic equations 68$y=ax^4$ 69$y=a(x-b)(x-c)(x-d)(x-e)$ 70$y=ax^4+cd^2 (c \ge 0)$ 71$y=ax^2(x-b)(x-c)$ 72$y=a(x-b)^2(x-c)^2$ 73$y=a(x-b)(x-c)^3$ 74 75## Simultaneous equations (linear) 76 77- **Unique solution** - lines intersect at point 78- **Infinitely many solutions** - lines are equal 79- **No solution** - lines are parallel 80 81### Solving $\protect\begin{cases}px + qy = a \\ rx + sy = b\protect\end{cases} \>$ for $\{0,1,\infty\}$ solutions 82 83where all coefficients are known except for one, and $a, b$ are known 84 851. Write as matrices: $\begin{bmatrix}p & q \\ r & s \end{bmatrix} 86\begin{bmatrix} x \\ y \end{bmatrix} 87 = 88\begin{bmatrix} a \\ b \end{bmatrix}$ 892. Find determinant of first matrix: $\Delta = ps-qr$ 903. Let $\Delta = 0$ for number of solutions $\ne 1$ 91 or let $\Delta \ne 0$ for one unique solution. 924. Solve determinant equation to find variable 93- *--- for infinite/no solutions: ---* 945. Substitute variable into both original equations 956. Rearrange equations so that LHS of each is the same 967. $\text{RHS}(1) = \text{RHS}(2) \implies (1)=(2) \> \forall x$ ($\infty$ solns) 97 $\text{RHS}(1) \ne \text{RHS}(2) \implies (1)\ne(2) \> \forall x$ (0 solns) 98 99\colorbox{cas}{On CAS:} Matrix $\rightarrow$ `det` 100 101### Solving $\protect\begin{cases}a_1 x + b_1 y + c_1 z = d_1 \\ 102a_2 x + b_2 y + c_2 z = d_2 \\ 103a_3 x + b_3 y + c_3 z = d_3\protect\end{cases}$ 104 105- Use elimination 106- Generate two new equations with only two variables 107- Rearrange & solve 108- Substitute one variable into another equation to find another variable 109- etc.