Arithmetic

factor = divisor

least common multiple

greatest common factor/divisor

results of division: 1. fraction 2. decimal 3. quotient with remainder

prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, and 29 …

prime factorization expressing a number as a product of factors that are prime numbers

composite number > 1 and not a prime number

fraction also called rational numbers

mixed number whole number + fraction

\[ 3^2 \] 3 is the base, 2 is the exponent

for a > 0:

\[a^0 = 1\] \[a^{-1} = \frac {1}{a}\]

square root operations

\[(\sqrt{a})^{2} = a\] \[\sqrt{a^{2}} = a\] \[\sqrt{a}\sqrt{b} = \sqrt{ab}\] \[\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}} \]

For odd-order roots, there is exactly one root for every number n, even when n is negative.

For even-order roots, there are exactly two roots for every positive number n and no roots for any negative number n.

irrational numbers decimals that do not terminate or repeat real numbers all irrational and rational numbers

division by 0 - undefined

triangle inequality \[ |a + b| \le |a| + |b|\]

\[|a||b| = |ab|\]

\[ a > 1, a^{2} > a\] \[ 0 < a < 1, b^{2} < b \]

proportion equation relating 2 ratios

\[ \frac {9}{12} = \frac {3}{4} \]

Algebra

identity A statement of equality between two algebraic expressions that is true for all possible values of the variables involved.

equation A statement of equality between two algebraic expressions that is true for only certain values of the variables involved.

Types of equations:

linear equation in one variable \[ 3x + 5 = -2 \]

linear equation in two variables \[ x - 3y = 10 \]

quadraticequationinonevariable \[ 20y^{2} + 6y - 17 = 0 \]

Rules of exponents \[ x^{-a} = \frac {1}{x^{a}}\] \[ (x^{a})(x^{b}) = x^{a+b} \] \[ \frac {x^{a}}{x^{b}} = x^{a-b} = \frac {1}{x^{b-a}} \] \[ x^{0} = 1 \] \[ (x^{a})(y^{a}) = (xy)^{a} \] \[ (\frac{x}{y})^{a} = \frac{x^{a}}{y^{a}}\] \[ (x^{a})^{b} = x^{ab}\]

common mistakes \[ x^{a}y^{b} \neq (xy)^{a+b}\] \[ (x^{a})^{b} \neq x^{a}y^{b} \] \[ (x + y)^{a} \neq x^{a} + y^{a} \] \[ -x^{2} \neq (-x)^{2} \] \[ \sqrt{x^{2} + y^{2}} \neq x + y \]

equivalent equations Two equations that have the same solutions.

to preserve equality of equations, either:

  1. add or subtract a constant to both sides
  2. multiply or divide a non-zero constant to both sides

linear equations consist of terms with a singe variable with or without a coefficient (raised to 1) and an optional constant

linear equations in 2 variables

system of equations Two equations with the same variables. simultaneous equations the equations in a system of equations.

To solve a system of two equations means to find an ordered pair of numbers that satisfies both equations in the system.

solving quadratic equations

\[ ax^{2} + bx + c = 0 \]

where: \[a \neq 0\]

quadratic formula solution to quadratic equation.

\[ x = \frac {-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

Inequalities

solve an inequality to find the set of all values of the variable that make the inequality true.

solution set of an inequality the set of values that solves an inequality.

equivalent inequalities Two inequalities that have the same solution set.

To solve an inequality

  1. add or subtract same constant to both sides. inequality is preserved.
  2. multiply or divide both sides with the same non-zero constant. inequality is preserved if constant is +. inequality is reversed if constant is -.

domain is assumed to be the set of all values of x for which f (x) is a real number.

Simple interest is based only on the initial deposit.

\[ V = P(1 + \frac {rt}{100})\]

where:
P - principal
V - final value
r - rate of interest
t - years/time priod

compound interest interest is added to the principal at regular time intervals, such as annually, quarterly, and monthly. After each compounding, interest is earned on the new principal.

Value V at the end of t years compounded n times per year

\[V = P(1 + \frac {r}{100n})^{nt}\]

where:
r - annual interest rate
P - principal

linear equation

\[ y = mx + b \]

where:
m - slope
b - y-intercept

Two lines are parallel if their slopes are equal. Two lines are perpendicular if their slopes are negative reciprocals of each other. slope of a line - rise over run \[ \frac {y_{2}-y_{1}}{x_{2}-x_{1}} \]

parabola

\[ y = ax^{2} + bx + c \]

where:
a, b, c - constants
\[ a \neq 0 \]

circle

\[ (x - a)^{2} + (y - b)^{2} = r^{2} \]

where:
a, b - coordinates of the center r - radius

parabola

\[ y = x^{2} - 2x - 3 \] \[ y = (x + 1)(x - 3) \] \[ x = -1, x = 3 \]

vertex is at midpoint between -1 and 3, 1
if x = 1, then y = 1 - 2 - 3 = -4

parabola

h(x) + c - shifts h(x) upward by c units h(x) - c - shifts h(x) downward by c units h(x + c) - shifts h(x) to the left by c units h(x - c) - shifts h(x) to the right by c units

parabola

ch(x) - if c > 1, stretch h(x) vertically by a factor of c ch(x) - if 0 < c < 1, shrink h(x) vertically by a factor of c

Geometry

congruent line segments line segments that have equal lengths congruent angles angles that have equal measures

opposite angles are congruent

opposite angles = vertical angles

convext polygon polygon where each interior angle < 180 degrees

sum of the measures of the interior angles of a polygon = (n-2)*180 degrees, n - no of sides

regular polygon all sides are congruent and all interior angles are congruent

equilateral triangle all sides are equal, all angles = 60 degrees

isosceles triangle at least 2 congruent sides

right triangle hypotenuse + 2 legs

similar triangles Two triangles that have the same shape but not necessarily the same size.
angles are congruent
the lengths of sides have the same ratio, called the scale factor of similarity

parallelogram A quadrilateral in which both pairs of opposite sides are parallel.

trapezoid A quadrilateral in which two opposite sides are parallel.

area of trapezoid

\[ A = \frac {1}{2}(b_{1} + b_{2})(h) \]

circles

congruent circles Two circles with equal radii.

chord Any line segment joining two points on the circle.

pi The ratio of the circumference C to the diameter d is the same for all circles.

\[ \frac {C}{d} = \pi \]

circumference

\[ C = 2\pi r \]

arc 2 points on a circle and all the points between them. Two points on a circle are always the endpoints of two arcs.

central angle A central angle of a circle is an angle with its vertex at the center of the circle.

measure of an arc is the measure of its central angle, which is the angle formed by two radii that connect the center of the circle to the two endpoints of the arc.

the ratio of the length of an arc to the circumference is equal to the ratio of the degree measure of the arc to 360 deg

\[ \frac {length of arc}{circumference} = \frac {degree measure of arc}{360^{o}} \]

area of a circle

\[ A = \pi r^{2} \]

sector is a region bounded by an arc of the circle and two radii.

the ratio of the area of a sector of a circle to the area of the entire circle is equal to the ratio of the degree measure of its arc to 360 deg

\[ \frac {A_{sector}}{A_{circle}} = \frac {sector^{o}}{360^{o}}\]

inscribed polygon all vertices of a polygon lie on the circle, or equivalently, the circle is circumscribed about the polygon.

circumscribed polygon if each side of the polygon is tangent to the circle, or equivalently, the circle is inscribed in the polygon.

concentric circles Two or more circles with the same center.

rectangular solid

\[ Volume V = l x w x h \]

\[ Surface Area A = 2(lw + lh + wh) \]

circular cylinder

** right circular cylinder** is a circular cylinder whose axis is perpendicular to its bases.

\[ Volume V = \pi r^{2}h \]

\[ Surface Area A = 2\pi r^{2} + 2\pi rh \]

Distributions of Data, Random Variables, and Probability Distributions

frequency, or count of a particular category or numerical value is the number of times that the category or value appears in the data.

frequency distribution is a table or graph that presents the categories or numerical values along with their associated frequencies. distribution curve = density curve = frequency curve

relative frequency of a category or a numerical value is the associated frequency divided by the total number of data.

graphs:

  1. bar graph
  2. segmented bar graph
  3. circle graphs
  4. histogram - intervals and their frequencies
  5. scatterplots
  6. time plot/time series
  7. boxplots / box-whisker plots

Measures of Central Tendency

  1. mean/average/arithmetic mean
  2. median
  3. mode

Measures of Position

  1. quartiles
  1. percentiles

Measures of Dispersion

dispersion the degree of “spread” of the data.

range = greatest number G - least number L

interquartile range A measure of dispersion that is not affected by outliers. Measures the spread of the middle half of the data.

\[ IR = Q_{3}-Q_{1} \]

boxplots

boxplot

standard deviation is a measure of spread that depends on each number in the list.

  1. sample standard deviation
  2. population standard deviation

In any group of data, most of the data are within about 3 standard deviations above or below the mean.

infinite sets Sets that are not finite.

finite set their members can be completely counted.

empty set is a subset of every set.

list is like a finite set. the members are ordered.

when the elements of a set are given, repetitions are not counted as additional elements and the order of the elements does not matter.

For any finite set S, the number of elements of S is denoted by |S|.

disjoint or mutually exclusive sets no elements in common.

inclusion-exclusion principle The number of elements in the union of two sets equals the sum of their individual numbers of elements minus the number of elements in their intersection.

multiplication principle k different possibilities for the first choice and m different possibilities for the second choice. each choice is independent of each other. There are km different possibilities for the pair of choices.

permutation order of objects.

number of permutations number of ways to order n objects. n!

permutations of n objects taken k at a time

\[ \frac {n!}{(n - k)!} \]

combinations does not consider order

(number of ways to select without order) x (number of ways to order) = (number of ways to select with order)

combinations of n objects taken k at a time / n choose k

\[ _{n}C_{k} = (^{n}_{k}) = \frac {n!}{k!(n -k)!}\]

n choose 0 = 1

n choose n = 1

Events E and F:

mutually exclusive Events that cannot occur at the same time.

independent events if the occurrence of either event does not affect the occurrence of the other.

P(E or F) = P(E) + P(F) - P(E and F)

P(E or F) = P(E) + P(F) if E and F are mutually exclusive.

P(E and F) = P(E)P(F) if E and F are independent.

If P(E) != 0 and P(F) != 0, then events E and F cannot be both mutually exclusive and independent.

freqdist

area under the curve = 1

probability distribution of a random variable is the same as the relative frequency distribution of the data (random variable chosen from a distribution of data)

expected value mean of a random variable

median halving point

mean balance point

Expected value of random variable X

\[ X_{mean} = \Sigma (Value * P(Value)) \]

discrete random variables - values are distrete points in the number line

in a histogram, the area of each bar is proportional to the probability represented by the bar

uniform distribution probability is distributed uniformly over all possible outcomes

The Normal Distribution

Properties

  1. mean ~ median ~ mode
  2. symmetrical about the mean
  3. 2/3 of data ~ within 1 standard deviation of the mean
  4. almost all of data within 2 standard deviations of the mean

continuous probability distribution region below a distribution curve

Standard Normal Distribution -> mean = 0 -> sd = 1

? transform a normal distribution to a standard normal distribution

Probability of X within 1 sd from mean = 0.683 Probability of X < 3 sd from mean = 0.0013