Methods - Semester 1

Simulatenous equations

Methods of solving:

• substitution (state one variable in terms of the other)
• subtraction (subtract one equation from the other, substitute resulting equation into the other)

Linear inequatlities

• Flip operator when multiplying / dividing by <0

Coordinate geometry

Regarding points $(x_1,y_1)(x_2y_2)$:

Midpoint: $m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})$

Distance: $d=\sqrt{(\Delta x)^2+(\Delta y)^2}$

Gradient: $m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}$

Line through points: $y={\Delta y \over \Delta x}(x+x_2)-y_2$

• parallel lines: $m_1=m_2$
• perpendicular lines: $m_1m_2=-1$

Polynomials

Binomial expansion: $(a+b)(c+d) = a(c+d) + b(c+d)$
Cubic expansion: $(a+b)^3=a^3+3a^2b+3ab^2+b^3$
Quartic expansion: $(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4$

Perfect square expansion: $(a+b)^2=a^2+2ab+b^2$
Difference of perfect squares: $a^2-b^2=(a+b)(a-b)$

Factorising quadratics: $x^2+bx+c = (x+e)(x-f)$
where $e \times f = c, \quad e+f = b$

Remainder theorem: if ${P({-\alpha \over \beta})}=0,$ then ${\beta x+ \alpha}$ is a factor of $P(x)$

Factor theorem: if $P(\alpha)=0,$ then $x-\alpha$ is a factor of $P(x)$

Completing the square (monic): $x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}$
Completing the square (non-monic): $ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}$

Quadratic formula: $x={{-b\pm\sqrt{b^2-4ac}}\over2a}$
Discriminant: $\Delta=b^2-4ac$

Solving $y_1=ax^2+bx+c_1, \quad y_2=mx+c_2$:

• $ax^2+(b-m)x+(c_1-c_2)=0$
• Solve for $x$, substitute into $y_2$

Axis of symmetry: $x={-b\over2a}$ (when written as $y=ax^2+bx+c$)

Determining quadratic rules:
$(1)\quad y=a(x-e)(x-f)$  two $x$-intercepts, one point
$(2)\quad y=a(x-h)^2+k$      turning point, one point
$(3)\quad y=ax^2+bx+c$        three points

Graphs

In general:

• turning point / centre point at $(h,k)$
• asymptotes at $x=h$ and $y=k$
• $a+$ dilates graph away from centre point
• $a<0$ reflects graph across $x=0$
• $h$ is the horizontal ($x$) shift, $k$ is the vertical ($y$) shift

Rectangular hyperbola: $\quad y={a\over x-h}+k$

• as $x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm$
• as $x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty$

Truncus:$\quad y={a\over (x-h)^2}+k$

• as $x \rightarrow \pm \infty, \quad y \rightarrow 0^+$
• as $x \rightarrow 0^{\pm}, \quad y \rightarrow \infty$

Square root:$\quad y=a\sqrt{x-h}+k$

• parabola rotated 90 degrees

Square root negative:$\quad y=a\sqrt{-(x-h)}+k$

• reflection of $y=\sqrt{x}$ across $y$-axis

Circle:$\quad (x-h)^2+(y-x)^2=r^2$

• factorised: $x^2+y^2-2hx-2ky+c=0$
• or: $y=\pm\sqrt{r^2-x^2}$

Semicircles - take +ve or -ve square root

• $y=\pm\sqrt{r^2-x^2}\quad$(top or bottom)
• $y=\pm\sqrt{r^2-y^2}\quad$(left or right)

Cubic: $\quad f(x)=a(x-h)^3+k$

• all cubics have >1 root

Inverse cubic: $\quad f^{-1}(x)=x^{1\over 3}$

• inverse of $\quad f(x)=a(x-h)^3+k$

Quartic: $\quad f(x)=a(x-h)^4+k$

Set notation

• set difference: $\quad A \setminus B=\{x:x\in A, x\notin B\}$
• interval notation: $(a,b)=\{x : a \lt x \lt b\},\quad$[a,b]={x : a \lte x \lte b}$

Functions

• function - one $y$ (image) value per $x$ (preimage)
• 1:1 function - unique $y$ for each $x$

Domain $\operatorname{dom}(f)$: set of all $x$ values in function

• maximal (implied) domain - largest domain for which the rule is defined
• restricted domain: $f(x)=\dots,\quad\operatorname{dom}(f)$

Range $\operatorname{ran}(f)$: set of all $y$ values in function

Piecewise functions: each domain has a corresponding equation

Inverse functions: $f^-1(x)=y$ if $f(y)=x,\quad$ for $x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)$

Methods of factorising cubics:

• extract common factor
• factor theorem
• polynomial division or equating coefficients
• sum or difference of two cubes
• quadratic formula

Polynomials: $P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0$

• degree ($n$) of $P(x)$ is the highest power of $x$

Bisection method

If $f(x)=0$ has a solution $\alpha$ between $[a_1,b_1]$:

• $f(a_1)\lt 0,\quad f(b_1)>0$
• calculate $f(c_1)=f({{a_1+b_1}\over 2})$
• if $f(c_1)\lt 0$, a root lies between $c_1$ and $b_1$
• if $f(c_1)\gt 0$, a root lies between $a_1$ and $c_1$

Matrices

• addition is only defined when dimensions are equal
• multiplication is only defined when columns in first = rows in second
• identity - equal to one
• inverse: $A^{-1}={1\over{ad-bc}}\times A\prime$
• determinant: $\det(A)=ad-bc$

Probability

• mutually exclusive: $\Pr(A\cap B)=0$
• independent: $\Pr(A|B)=\Pr(A)$
• addition rule: $\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)$
• multiplication rule: $\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)$
• law of total probability: $\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)$
• conditional probability: $\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}$

Combinatorics

• Arrangements of $n$ in $r$ is given by $n!\over{(n-r)!}$
• Combinations of $n$ in $r$ is given by $n! \over{r!(n-r)!}$