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44 <h1>Methods - Semester 1</h1>
45<h2>Simulatenous equations</h2>
46<p>Methods of solving:</p>
47<ul>
48<li>substitution (state one variable in terms of the other)</li>
49<li>subtraction (subtract one equation from the other, substitute resulting equation into the other)</li>
50</ul>
51<h2>Linear inequatlities</h2>
52<ul>
53<li>Flip operator when multiplying / dividing by <0</li>
54</ul>
55<h2>Coordinate geometry</h2>
56<p>Regarding points <span class="math"><script type="math/tex">(x_1,y_1)(x_2y_2)</script></span>:</p>
57<p><strong>Midpoint:</strong> <span class="math"><script type="math/tex">m=({{\Sigma x}\over 2}, {{\Sigma y}\over 2})</script></span></p>
58<p><strong>Distance:</strong> <span class="math"><script type="math/tex">d=\sqrt{(\Delta x)^2+(\Delta y)^2}</script></span></p>
59<p><strong>Gradient:</strong> <span class="math"><script type="math/tex">m={\operatorname{rise}\over\operatorname{run}}={\Delta y \over \Delta x}</script></span></p>
60<p><strong>Line through points:</strong> <span class="math"><script type="math/tex">y={\Delta y \over \Delta x}(x+x_2)-y_2</script></span></p>
61<ul>
62<li>parallel lines: <span class="math"><script type="math/tex">m_1=m_2</script></span></li>
63<li>perpendicular lines: <span class="math"><script type="math/tex">m_1m_2=-1</script></span></li>
64</ul>
65<h2>Polynomials</h2>
66<p><strong>Binomial expansion:</strong> <span class="math"><script type="math/tex">(a+b)(c+d) = a(c+d) + b(c+d)</script></span><br>
67<strong>Cubic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^3=a^3+3a^2b+3ab^2+b^3</script></span><br>
68<strong>Quartic expansion:</strong> <span class="math"><script type="math/tex">(a+b)^4=a^4+4a^3b+6a^2b^2+4ab^3+b^4</script></span></p>
69<p><strong>Perfect square expansion:</strong> <span class="math"><script type="math/tex">(a+b)^2=a^2+2ab+b^2</script></span><br>
70<strong>Difference of perfect squares:</strong> <span class="math"><script type="math/tex">a^2-b^2=(a+b)(a-b)</script></span></p>
71<p><strong>Factorising quadratics:</strong> <span class="math"><script type="math/tex">x^2+bx+c = (x+e)(x-f)</script></span><br>
72where <span class="math"><script type="math/tex">e \times f = c, \quad e+f = b</script></span></p>
73<p><strong>Remainder theorem:</strong> if <span class="math"><script type="math/tex">{P({-\alpha \over \beta})}=0,</script></span> then <span class="math"><script type="math/tex">{\beta x+ \alpha}</script></span> is a factor of <span class="math"><script type="math/tex">P(x)</script></span></p>
74<p><strong>Factor theorem:</strong> if <span class="math"><script type="math/tex">P(\alpha)=0,$then</script></span>x-\alpha$ is a factor of $P(x)</p>
75<p><strong>Completing the square (monic):</strong> <span class="math"><script type="math/tex">x^2+bx+c=(x+{b\over2})^2+c-{b^2\over4}</script></span><br>
76<strong>Completing the square (non-monic):</strong> <span class="math"><script type="math/tex">ax^2+bx+c=a(x-{b\over2a})^2+c-{b^2\over4a}</script></span></p>
77<p><strong>Quadratic formula:</strong> <span class="math"><script type="math/tex">x={{-b\pm\sqrt{b^2-4ac}}\over2a}</script></span><br>
78<strong>Discriminant:</strong> <span class="math"><script type="math/tex">\Delta=b^2-4ac</script></span></p>
79<p><strong>Solving</strong> <span class="math"><script type="math/tex">y_1=ax^2+bx+c_1, \quad y_2=mx+c_2</script></span>:</p>
80<ul>
81<li><span class="math"><script type="math/tex">ax^2+(b-m)x+(c_1-c_2)=0</script></span></li>
82<li>Solve for <span class="math"><script type="math/tex">x</script></span>, substitute into <span class="math"><script type="math/tex">y_2</script></span></li>
83</ul>
84<p><strong>Axis of symmetry:</strong> <span class="math"><script type="math/tex">x={-b\over2a}</script></span> (when written as <span class="math"><script type="math/tex">y=ax^2+bx+c</script></span>)</p>
85<p><strong>Determining quadratic rules:</strong><br>
86<span class="math"><script type="math/tex">(1)\quad y=a(x-e)(x-f)</script></span> two <span class="math"><script type="math/tex">x</script></span>-intercepts, one point<br>
87<span class="math"><script type="math/tex">(2)\quad y=a(x-h)^2+k</script></span> turning point, one point<br>
88<span class="math"><script type="math/tex">(3)\quad y=ax^2+bx+c</script></span> three points</p>
89<h2>Graphs</h2>
90<p>In general:</p>
91<ul>
92<li>turning point / centre point at <span class="math"><script type="math/tex">(h,k)</script></span></li>
93<li>asymptotes at <span class="math"><script type="math/tex">x=h</script></span> and <span class="math"><script type="math/tex">y=k</script></span></li>
94<li><span class="math"><script type="math/tex">a+</script></span> dilates graph away from centre point</li>
95<li><span class="math"><script type="math/tex">a<0</script></span> reflects graph across <span class="math"><script type="math/tex">x=0</script></span></li>
96<li><span class="math"><script type="math/tex">h</script></span> is the horizontal (<span class="math"><script type="math/tex">x</script></span>) shift, <span class="math"><script type="math/tex">k</script></span> is the vertical (<span class="math"><script type="math/tex">y</script></span>) shift</li>
97</ul>
98<p><strong>Rectangular hyperbola:</strong> <span class="math"><script type="math/tex">\quad y={a\over x-h}+k</script></span></p>
99<ul>
100<li>as <span class="math"><script type="math/tex">x\rightarrow \pm\infty, \quad y \rightarrow 0^\pm</script></span></li>
101<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \pm \infty</script></span></li>
102</ul>
103<p><strong>Truncus:</strong><span class="math"><script type="math/tex">\quad y={a\over (x-h)^2}+k</script></span></p>
104<ul>
105<li>as <span class="math"><script type="math/tex">x \rightarrow \pm \infty, \quad y \rightarrow 0^+</script></span></li>
106<li>as <span class="math"><script type="math/tex">x \rightarrow 0^{\pm}, \quad y \rightarrow \infty</script></span></li>
107</ul>
108<p><strong>Square root:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{x-h}+k</script></span></p>
109<ul>
110<li>parabola rotated 90 degrees</li>
111</ul>
112<p><strong>Square root negative:</strong><span class="math"><script type="math/tex">\quad y=a\sqrt{-(x-h)}+k</script></span></p>
113<ul>
114<li>reflection of <span class="math"><script type="math/tex">y=\sqrt{x}</script></span> across <span class="math"><script type="math/tex">y</script></span>-axis</li>
115</ul>
116<p><strong>Circle:</strong><span class="math"><script type="math/tex">\quad (x-h)^2+(y-x)^2=r^2</script></span></p>
117<ul>
118<li>factorised: <span class="math"><script type="math/tex">x^2+y^2-2hx-2ky+c=0</script></span></li>
119<li>or: <span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}</script></span></li>
120</ul>
121<p><strong>Semicircles</strong> - take +ve or -ve square root</p>
122<ul>
123<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-x^2}\quad</script></span>(top or bottom)</li>
124<li><span class="math"><script type="math/tex">y=\pm\sqrt{r^2-y^2}\quad</script></span>(left or right)</li>
125</ul>
126<p><strong>Cubic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></p>
127<ul>
128<li>all cubics have >1 root</li>
129</ul>
130<p><strong>Inverse cubic:</strong> <span class="math"><script type="math/tex">\quad f^{-1}(x)=x^{1\over 3}</script></span></p>
131<ul>
132<li>inverse of <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^3+k</script></span></li>
133</ul>
134<p><strong>Quartic:</strong> <span class="math"><script type="math/tex">\quad f(x)=a(x-h)^4+k</script></span></p>
135<h2>Set notation</h2>
136<ul>
137<li>set difference: <span class="math"><script type="math/tex">\quad A \setminus B=\{x:x\in A, x\notin B\}</script></span></li>
138<li>interval notation: <span class="math"><script type="math/tex">(a,b)=\{x : a \lt x \lt b\},\quad</script></span>[a,b]={x : a \lte x \lte b}$</li>
139</ul>
140<h2>Functions</h2>
141<span class="math"><script type="math/tex; mode=display">f:\operatorname{dom}(f) \rightarrow \mathbb{R},\quad f(x)=\dots
142</script></span>
143<ul>
144<li>function - one <span class="math"><script type="math/tex">y</script></span> (image) value per <span class="math"><script type="math/tex">x</script></span> (preimage)</li>
145<li>1:1 function - unique <span class="math"><script type="math/tex">y</script></span> for each <span class="math"><script type="math/tex">x</script></span></li>
146</ul>
147<p><strong>Domain <span class="math"><script type="math/tex">\operatorname{dom}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">x</script></span> values in function</p>
148<ul>
149<li>maximal (implied) domain - largest domain for which the rule is defined</li>
150<li>restricted domain: <span class="math"><script type="math/tex">f(x)=\dots,\quad\operatorname{dom}(f)</script></span></li>
151</ul>
152<p><strong>Range <span class="math"><script type="math/tex">\operatorname{ran}(f)</script></span>:</strong> set of all <span class="math"><script type="math/tex">y</script></span> values in function</p>
153<p><strong>Piecewise functions:</strong> each domain has a corresponding equation</p>
154<p><strong>Inverse functions:</strong> <span class="math"><script type="math/tex">f^-1(x)=y</script></span> if <span class="math"><script type="math/tex">f(y)=x,\quad</script></span> for <span class="math"><script type="math/tex">x\in \operatorname{ran}(f), y\in \operatorname{dom}(f)</script></span></p>
155<p><strong>Methods of factorising cubics:</strong></p>
156<ul>
157<li>extract common factor</li>
158<li>factor theorem</li>
159<li>polynomial division or equating coefficients</li>
160<li>sum or difference of two cubes</li>
161<li>quadratic formula</li>
162</ul>
163<p><strong>Polynomials:</strong> <span class="math"><script type="math/tex">P(x)=a_nx^n+a_{n-1}x^{n-1}+\dots a_1x+a_0</script></span></p>
164<ul>
165<li>degree (<span class="math"><script type="math/tex">n</script></span>) of <span class="math"><script type="math/tex">P(x)</script></span> is the highest power of <span class="math"><script type="math/tex">x</script></span></li>
166</ul>
167<h2>Bisection method</h2>
168<p>If <span class="math"><script type="math/tex">f(x)=0</script></span> has a solution <span class="math"><script type="math/tex">\alpha</script></span> between <span class="math"><script type="math/tex">[a_1,b_1]</script></span>:</p>
169<ul>
170<li><span class="math"><script type="math/tex">f(a_1)\lt 0,\quad f(b_1)>0</script></span></li>
171<li>calculate <span class="math"><script type="math/tex">f(c_1)=f({{a_1+b_1}\over 2})</script></span></li>
172<li>if <span class="math"><script type="math/tex">f(c_1)\lt 0</script></span>, a root lies between <span class="math"><script type="math/tex">c_1</script></span> and <span class="math"><script type="math/tex">b_1</script></span></li>
173<li>if <span class="math"><script type="math/tex">f(c_1)\gt 0</script></span>, a root lies between <span class="math"><script type="math/tex">a_1</script></span> and <span class="math"><script type="math/tex">c_1</script></span></li>
174</ul>
175<h2>Matrices</h2>
176<ul>
177<li>addition is only defined when dimensions are equal</li>
178<li>multiplication is only defined when columns in first = rows in second</li>
179<li>identity - equal to one</li>
180<li>inverse: <span class="math"><script type="math/tex">A^{-1}={1\over{ad-bc}}\times A\prime</script></span></li>
181<li>determinant: <span class="math"><script type="math/tex">\det(A)=ad-bc</script></span></li>
182</ul>
183<h2>Probability</h2>
184<ul>
185<li>mutually exclusive: <span class="math"><script type="math/tex">\Pr(A\cap B)=0</script></span></li>
186<li>independent: <span class="math"><script type="math/tex">\Pr(A|B)=\Pr(A)</script></span></li>
187<li>addition rule: <span class="math"><script type="math/tex">\Pr(A\cup B)=\Pr(A)+\Pr(B)-\Pr(A\cap B)</script></span></li>
188<li>multiplication rule: <span class="math"><script type="math/tex">\Pr(A\cap B)=\Pr(A|B)\times \Pr(B)</script></span></li>
189<li>law of total probability: <span class="math"><script type="math/tex">\Pr(A)=\Pr(A|B)\Pr(B)+\Pr(A|B\prime)\Pr(B\prime)</script></span></li>
190<li>conditional probability: <span class="math"><script type="math/tex">\Pr(A|B)={{\Pr(A\cap B)\over\Pr(B)}}</script></span></li>
191</ul>
192<h2>Combinatorics</h2>
193<ul>
194<li>Arrangements of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n!\over{(n-r)!}</script></span></li>
195<li>Combinations of <span class="math"><script type="math/tex">n</script></span> in <span class="math"><script type="math/tex">r</script></span> is given by <span class="math"><script type="math/tex">n! \over{r!(n-r)!}</script></span></li>
196</ul>
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