Terms and Formulas

Aida Haghighi

Terms and Formulas

Aida Haghighi

Term

Formula

Explanation

Ratio	a : b
Proportion	$\frac{a}{b}=\frac{c}{d}$	An equation with a ratio (or rate) on two sides, in which the two ratios are equal.
Cross-product rule	$\frac{a}{b} = \frac{c}{d} \quad \text{equals} \quad a \times d = c \times b$	Multiplying along two diagonals and solving for the unknown.
Percent proportion method	$\frac{\text{Part ("is")}}{\text{Whole ("of")}} = \frac{\%}{100}$
Percent increase	= $\frac{New\;value-Original\;value}{Original\;value}$
Percent decrease	= $\frac{Original\;value-New\;value}{Original\;value}$
Linear equation: slope-intercept form	y = mx + b	m is the slope and b is the constant.
Slope of a linear equation	$m = \frac{y_2 - y_1}{x_2 - x_1}$	$(x_1, y_1) \text{ and } (x_2, y_2) \text{ are two points.}$
Linear equation: standard form	Ax + By = C
Slope of a linear equation in standard form	$m = -\frac{A}{B}$
Linear equation: Point-slope formula	$y-y_1=m(x-x_1)$	$(x_1, y_1) \text{ is a point.}$
Average rate of change	$\frac{Change\;of\;Output}{Change\;of\;Input}=\frac{\Delta y}{\Delta x}=\frac{y_2-y_1}{ x_2-x_1}$	The average rate of change between two input values is the total change of the function values (output values) divided by the change in the input values.
Quadratic function: standard form	$f(x)=ax^2+bx+c$
Quadratic function: vertex form	$f(x)=a(x-h)^2+k$	$(h, k) \text{ is the vertex point.}$
Vertex of a quadratic function	$h=-\frac{b}{2a}$ , $k=f(h)=f(\frac{-b}{2a})$
Quadratic formula	$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}}$	It gives the horizontal intercepts.
Exponential growth or decay function	$f(x)=a(1+r)^x$ or $f(x)=ab^x$ where $b=1+r$	a is the initial or starting value, r is the percent growth or decay rate, and b is the growth factor. $\text{Note: if we have a decay rate, } 1 + r \text{ should be changed to } 1 - r \text{.}$
Compound Interest Formula	$A(t)=a(1+\frac{r}{k})^{kt}$	r is the annual percentage rate (APR), also called the nominal rate, and k is the number of compounding periods in one year
Continuous Growth Formula	f(x) = $ae^{rx}$	$\text{Note: if we have a decay rate, } r \text{ should be changed to } - r \text{.}$
Conversion to log form	b^a = c is equivalent to the statement log_b(c) = a .
Properties of Logs: Inverse Properties	log_b(b^x) = x b^log_bx = x
Properties of Logs: Exponent Property	log_b(A^r) = rlog_b(A)
Properties of Logs: Change of Base	$\log_{b}(A)=\frac{\log_{c}(A)}{\log_{c}(b)}$
Sum of Logs Property	$\log_{b}(A)+\log_{b}(C)=\log_b(AC)$
Difference of Logs Property	$\log_{b}(A)-log_{b}(C)=\log_{b}(\frac{A}{C})$
Half-Life (Based on standard exponential function)	$\frac{1}{2}=b^t$
Half-Life (Based on continuous change function)	$\frac{1}{2}=e^{rt}$
Doubling Time (Based on standard exponential function)	2 = b^t
Half-Life (Based on continuous change function)	$2=e^{rt}$
Subset	$A\subseteq B$	set A is a subset of a set B if every member of A is also a member of B.
Union of two sets	$A\cup B$	The set of all elements that are either in A or in B, or in both.
Intersection of two sets	$A\cap B$	The set of all elements that are common to both sets A and B.
Complement of a set	$A^c$	The set consists of elements in the universal set U that are not in A.
Permutations	nPr = $\frac{n!}{(n-r)!}$	The Number of Permutations of n Objects Taken r at a Time.
Circular Permutations	(n − 1)!	The number of permutations of n elements in a circle.
Permutations with Similar Elements	$\frac{n!}{r_1!r_2!...r_k!}$	The number of permutations of n elements taken n at a time, with r₁ elements of one kind, r₂ elements of another kind, and so on.
Combinations	nCr = $\frac{n!}{(n-r)!r!}$	The Number of Combinations of n Objects Taken r at a Time
Probability addition rule	$P(E\cup F)=P(E)+P(F)-P(E\cap F)$	The probability of the union of two events
The complement rule	$P(E^c)= 1-P(E)$
Conditional probability	P(E \| F) = $\frac{P(E\cap F)}{P(F)}$	The probability of E given F
Independence test	$P(E\cap F) = P(E)\,P(F)$	Two Events E and F are independent.
Binomial Probability Theorem	P(n, k; p) = nCkp^kq^{n – k}	p denotes the probability of success and q = (1 − p) the probability of failure.
Bayes’ Formula	$P(A_i\,\|\, E) = \frac{P(A_i)\, P(E\,\|\, A_i)}{P(A_1)\, P(E\,\|\, A_1)+P(A_2)\, P(E\,\|\, A_2)+\,\cdots\,+P(A_n)\, P(E\,\|\, A_n)}$	Let S be a sample space that is divided into n partitions, A₁, A₂, . . . A_nand E is any event in S.
Expected Value	Expected Value = x₁p(x₁) + x₂p(x₂) + x₃p(x₃) + ··· + x_np(x_n)	$p(x_1) \text{ is the probability of } x_1, \quad p(x_2) \text{ is the probability of } x_2, \quad \text{and more}$
Markov Chains	$\left[\begin{array}{cc} x_1 & x_2\end{array}\right] \left[\begin{array}{cc} y_1 & y_3 \\y_2 & y_4\end{array}\right]=\left[\begin{array}{cc} (x_1)(y_1) + (x_2)(y_2) & (x_1)(y_3) + (x_2)(y_4)\end{array}\right]$
Equilibrium vector	ET = E $E=\left[\begin{array}{cc} e & 1-e\end{array}\right]$	The system is in steady-state or state of equilibrium in the long run. T is the transition matrix and E is the equilibrium vector.