1# Methods - Semester 1 2 3## Simulatenous equations 4Methods of solving: 5- substitution (state one variable in terms of the other) 6- subtraction (subtract one equation from the other, substitute resulting equation into the other) 7 8## Linear inequatlities 9 10- Flip operator when multiplying / dividing by <0 11 12## Coordinate geometry 13 14Regarding points $(x_1,y_1)(x_2y_2)$: 15 16**Midpoint:** $m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})$ 17 18**Distance:** $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$ 19 20**Gradient:** $m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}$ 21 22**Line through points:** $y={\Delta y \over \Delta x}(x+x_2)-y_2$ 23 24- parallel lines: $m_1=m_2$ 25- perpendicular lines: $m_1m_2=-1$ 26 27## Polynomials 28 29**Binomial expansion:** $(a+b)(c+d) = a(c+d) + b(c+d)$ 30**Cubic expansion:** $(a+b)^3=a^3+3a^2b+3ab^2+b^3$ 31**Quartic expansion:** $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$ 32 33**Perfect square expansion:** $(a+b)^2=a^2+2ab+b^2$ 34**Difference of perfect squares:** $a^2-b^2=(a+b)(a-b)$ 35 36**Factorising quadratics:** $x^2+bx+c = (x+e)(x-f)$ 37where $e \times f = c, \quad e+f = b$ 38 39**Remainder theorem:** if ${P({-\alpha \over \beta})}=0,$ then ${\beta x+ \alpha}$ is a factor of $P(x)$ 40 41**Factor theorem:** if $P(\alpha)=0,$ then $x-\alpha$ is a factor of $P(x)$ 42 43**Completing the square (monic):** $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$ 44**Completing the square (non-monic):** $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$ 45 46**Quadratic formula:** $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$ 47**Discriminant:** $\Delta=b^2-4ac$ 48 49**Solving** $y_1=ax^2+bx+c_1, \quad y_2=mx+c_2$: 50- $ax^2+(b-m)x+(c_1-c_2)=0$ 51- Solve for $x$, substitute into $y_2$ 52 53**Axis of symmetry:** $x={-b\over2a}$ (when written as $y=ax^2+bx+c$) 54 55**Determining quadratic rules:** 56$(1)\quad y=a(x-e)(x-f)$ two $x$-intercepts, one point 57$(2)\quad y=a(x-h)^2+k$ turning point, one point 58$(3)\quad y=ax^2+bx+c$ three points 59 60## Graphs 61 62In general: 63- turning point / centre point at $(h,k)$ 64- asymptotes at $x=h$ and $y=k$ 65- $a+$ dilates graph away from centre point 66- $a<0$ reflects graph across $x=0$ 67- $h$ is the horizontal ($x$) shift, $k$ is the vertical ($y$) shift 68 69**Rectangular hyperbola:** $\quad y={a\over x-h}+k$ 70- as $x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm$ 71- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty$ 72 73**Truncus:**$\quad y={a\over (x-h)^2}+k$ 74- as $x \rightarrow \pm \infty, \quad y \rightarrow 0^+$ 75- as $x \rightarrow 0^{\pm}, \quad y \rightarrow \infty$ 76 77**Square root:**$\quad y=a\sqrt{x-h}+k$ 78- parabola rotated 90 degrees 79 80**Square root negative:**$\quad y=a\sqrt{-(x-h)}+k$ 81- reflection of $y=\sqrt{x}$ across $y$-axis 82 83**Circle:**$\quad (x-h)^2+(y-x)^2=r^2$ 84- factorised: $x^2+y^2-2hx-2ky+c=0$ 85- or: $y=\pm\sqrt{r^2-x^2}$ 86 87**Semicircles** - take +ve or -ve square root 88- $y=\pm\sqrt{r^2-x^2}\quad$(top or bottom) 89- $y=\pm\sqrt{r^2-y^2}\quad$(left or right) 90 91**Cubic:** $\quad f(x)=a(x-h)^3+k$ 92- all cubics have >1 root 93 94**Inverse cubic:** $\quad f^{-1}(x)=x^{1\over 3}$ 95- inverse of $\quad f(x)=a(x-h)^3+k$ 96 97**Quartic:** $\quad f(x)=a(x-h)^4+k$ 98 99 100## Set notation 101 102- set difference: $\quad A \setminus B=\{x:x\in A, x\notin B\}$ 103- interval notation: $(a,b)=\{x : a \lt x \lt b\},\quad$[a,b]=\{x : a \lte x \lte b\}$ 104 105## Functions 106$$f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots$$ 107- function - one $y$ (image) value per $x$ (preimage) 108-1:1 function - unique $y$ for each $x$ ($\sin x$ is not 1:1, $x^3$ is) 109 110**Domain $\operatorname{dom}(f)$:** set of all $x$ values in function 111- maximal (implied) domain - largest domain for which the rule is defined 112- restricted domain: $f(x)=\dots,\quad\operatorname{dom}(f)$ 113 114**Range $\operatorname{ran}(f)$:** set of all $y$ values in function 115 116**Piecewise functions:** each domain has a corresponding equation 117 118**Inverse functions:** $f^-1(x)=y$ if $f(y)=x,\quad$ for $x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)$ 119 120**Methods of factorising cubics:** 121- extract common factor 122- factor theorem 123- polynomial division or equating coefficients 124- sum or difference of two cubes 125- quadratic formula 126 127**Polynomials:** $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$ 128- degree ($n$) of $P(x)$ is the highest power of $x$ 129 130## Bisection method 131If $f(x)=0$ has a solution $\alpha$ between $[a_1,b_1]$: 132- $f(a_1)\lt 0,\quad f(b_1)>0$ 133- calculate $f(c_1)=f({{a_1+b_1}\over 2})$ 134- if $f(c_1)\lt 0$, a root lies between $c_1$ and $b_1$ 135- if $f(c_1)\gt 0$, a root lies between $a_1$ and $c_1$ 136 137## Matrices 138- addition is only defined when dimensions are equal 139- multiplication is only defined when columns in first = rows in second 140- identity - equal to one 141- inverse: $A^{-1}={1\over{ad-bc}}\times A\prime$ 142- determinant: $\det(A)=ad-bc$ 143 144## Probability 145- mutually exclusive: $\Pr(A\cap B)=0$ 146- independent: $\Pr(A|B)=\Pr(A)$ 147- addition rule: $\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)$ 148- multiplication rule: $\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)$ 149- law of total probability: $\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)$ 150- conditional probability: $\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}$ 151 152## Combinatorics 153 154- Arrangements of $n$ in $r$ is given by $n!\over{(n-r)!}$ 155- Combinations of $n$ in $r$ is given by $n! \over{r!(n-r)!}$