Math- Things to Know

Algebra

linear equation and inequalities
1. inequalities
  1. positive coefficient of x = maintain direction
  2. negative coefficient of x = reverse direction
linear equation word problems
1. $s l o p e = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}$
2. slope-intercept form $$ y = mx + b $$
Interpreting linear problems
graphs of linear equations and inequalities
systems of linear equations
1. steps:
  1. rewrite both in slope-intercept form
  2. compare $m$ and $b$ value
  3. if $m_{1} \neq m_{2}$ , one solution
  4. if $m_{1} = m_{2}$ and $b_{1} \neq b_{2}$ , no solution
  5. if $m_{1} = m_{2}$ and $b_{1} = b_{2}$ , infinitely many solution
linear inequality word problems

word	word/syntax
more than, greater that, higher than	$>$
less than, lower than	$<$
greater than or equal to, at least, no less than	$\geq$
less than or equal to, at most, no more than	$\leq$
least, lowest, minimum	smallest value
greatest, highest, maximum	largest value
a possible	any satisfiable value

graphs of linear systems and inequalities
1. linear inequalities in slope-intercept form
  1. $y > m x + b$ : shade above the line
  2. $y < m x + b$ : shade below the line
2. $>$ or $<$ : DON'T include points in solution set
3. $\geq$ or $\leq$ : DO include points in solution set

Advanced Math

Factoring quadratic and polynomial expressions
1. ${(a + b)}^{2} = a^{2} + 2 a b + b^{2}$
2. ${(a - b)}^{2} = a^{2} - 2 a b + b^{2}$
3. $a^{2} - b^{2} = (a + b) (a - b)$
Radicals and rational exponents
1. $a x^{n} \pm b x^{n} = (a \pm b) x^{n}$
2. $a x^{m} \cdot b x^{n} = a b \cdot x^{m + n}$
3. $x^{n} \cdot y^{n} = {(x y)}^{n}$
4. $\frac{x^{n}}{y^{n}} = {(\frac{x}{y})}^{n}$
5. many more...
Operations with polynomials
1. FOIL for multiplying two binomials: First, Outer, Inner, Last
Operations with rational expressions
non-linear functions
isolating quantities
solving quadratic equations
1. form: $a x^{2} + b x + c = 0$ , then $x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$
2. $b^{2} - 4 a c > 0$ ; 2 unique real solutions
3. $b^{2} - 4 a c = 0$ ; 1 unique real solutions
4. $b^{2} - 4 a c < 0$ ; NO unique real solutions
linear and quadratic systems
1. One linear equation and One quadratic equation
radical, rational and absolute value equation
1. radical operator $\to$ $\sqrt{a}$
2. rational operator $\to$ $\frac{a}{b}$
3. absolute value operator $\to$ $| a |$
quadratic and exponential word problems
1. height vs. time

\begin{array}{r} h (t) = - a t^{2} + b t + c \end{array}

$h (t)$ $\to$ height at a time
$t$ $\to$ time
$- a t^{2}$ $\to$ gravity effect
$b t$ $\to$ initial upward velocity
$c$ $\to$ initial height when $t = 0$
$a$ $\to$ $\frac{1}{2}$ of acceleration due to gravity. as acceleration due to gravity is $32 f t / s^{2}$ or $9.8 m / s^{2}$ , $a$ = $16 \frac{f t}{s^{2}}$ or $4.9 \frac{m}{s^{2}}$
maximum height = $\frac{b}{2 a}$
2. changing population

\begin{array}{r} P (t) = P_{0} \cdot r t \\ P (t) \to p o p u l a t i o n a f t e r t i m e \\ t \to t i m e e l a p s e d \\ P_{0} \to i n i t i a l p o p u l a t i o n \\ r \to h o w p o p u l a t i o n i s c h a n g i n g \\ r > 1; p o p u l a t i o n g r o w i n g \\ 0 < r < 1; p o p u l a t i o n d e c l i n i n g \end{array}

3. compounding interest

\begin{array}{r} P (t) = P_{0} {(1 + r)}^{t} \\ P_{0} \to i n i t i a l a m o u n t o f m o n e y \\ r \to i n t e r e s t r a t e \\ t \to t i m e e l a p s e d \end{array}

4. time conversion
1. only exponents $t$ change
2. small unit $\to$ large unit; multiply
3. large unit $\to$ small unit; divide

quadratic graphs: parabolas
1. vertex = max/ min $y$ -values
  1. upward = minimum vertex ; $a > 0$ [ $a$ is the coefficient of $x^{2}$ ]
  2. downward = maximum vertex ; $a < 0$
2. vertex form of quadratic equation

\begin{array}{r} y = a {(x - h)}^{2} + k \\ w h e r e, \\ v e r t e x p o i n t i s (h, k) \end{array}

3. x-intercept = zero = root ; same thing