Math Terms

Arithmetic sequence

a sequence in which the difference between each term is constant, There is a constant difference between two consecutive terms

Geometric sequence

a sequence in which the ratio between two consecutive terms is constant

Infinite sequence

a sequence that consists of an infinite amount of terms

Finite sequence

a sequence with a domain that consists of the first n positive integers only

Cofunction

a trigonometric function whose value for the complement of an angle is equal to the value of a given trigonometric function of the angle itself. identities: sinø = cos(90º-ø), cosø = sin(90º-ø), tanø = cot(90º-ø), cotø = tan(90º-ø), cscø = sec(90º-ø), secø = csc(90º-ø)

Imaginary

a value that gives a negative result when squared (e.g. the square root of a negative number)

Independent

a variable (often denoted by x ) whose variation does not depend on that of another

Dependent

a variable (often denoted by y ) whose value depends on that of another.

Polynomials

an expression consisting of a sum of terms that include one or more variables raised to powers and multiplied by a coefficient

Zeros

an input value of a function which produces an output value of zero. Also called roots, Where the x value is zero

closed interval

an interval that does contain its endpoints [indicated with hard brackets, cannot be used with infinities]

open interval

an interval that does not contain its endpoints (indicated with soft brackets, always used with infinities)

Radians

angle made when the radius is wrapped around the circle, unit of measure for angles

Complementary

angles that add up to 90 degrees

Supplementary

angles that as up to 180 degrees

Binomial theorem

any binomial (x + y)^n can be expanded into n+1 terms

Continuous function

any value of x will give us a corresponding value of y (no holes)

convergence

as the series (or function) increases or decreases, it approaches a limit closer and closer. In a geometric series, the series will either converge or diverge: Convergence is when the graphed function gets closer to zero, and it can be computed, The absolute value of the ratio must be less than one to converge. In other words, as the x values go to infinity, the y values get closer to zero

Divergence

as the series (or function) increases or decreases, it moves farther and farther away from a limit, If the absolute value of the ratio in a geometric sequence is greater than one, it will diverge, meaning it will branch off and go on for infinity, Cannot be computed, In other words, as x approaches infinity, the y values approach infinity

Decreasing function

as x approaches positive infinity, y approaches negative infinity (y values decrease as x values increase)

Increasing function

as x approaches positive infinity, y approaches positive infinity (y values decrease as x values increase)

Constant function

as x approaches positive infinity, y remains constant, Horizontal function

Trigonometry

branch of mathematics that studies the relationships between lengths and angles of triangles

The unit circle

circle centered at the origin with a radius of 1 which corresponds to the hypotenuse of the right triangle for an angle

Cartesian coordinates

define a point in space with coordinates (x,y) where you start at the origin, move x units horizontally and then y units vertically (there is only one form of coordinates for every point)

Polar coordinates

describe the position of a point in terms of a distance from a fixed point (origin) and an angle measured from the x-axis (there is an infinite number of coordinates for a given point based on how you approach the point), conversion to cartesian coordinates: (R•cosø, R•sinø)

Leading term analysis

determining traits about a graph based on the term in a polynomial with the highest power of the variable. examples: positive even leading term, negative even leading term, positive odd leading term, negative odd leading term. can tell you end behavior, concavity, turning points

midpoint

divides a line segment in two equal segments (bisects into equal halves), formula = [(x1 + x2)/2 , (y1 + y2)/2]

Tau

represents 2 pi, or one full rotation of a circle

Pythagoreans identity

sin2ø + cos2ø = 1, can divide by either sin2ø or cos2ø to reach the other values (e.g. tan, cot, csc, sec)

concave down

slope decreases (looks like an upside-down bowl), a tangent line to the graph at any point lies above the graph

concave up

slope increases (looks like a right-side up bowl), a tangent line to the graph at any point lies below the graph

Lateral

surface area of an object

Angle

the amount of turn between two rays connected by a vertex

End behavior

the behavior of the graph ƒ(x) as x approaches positive infinity or negative infinity

Maximum

the height of the function at a certain point is greater than the height anywhere else at that interval (≈ a hill: highest point in a certain section of a graph)

Minimum

the height of the function at a certain point that is less than the height anywhere else at that interval (≈ a valley: lowest point in a certain section of a graph)

Linear factorization theorem

the highest degree for a polynomial has that many zeros, allows you to factor a polynomial to the point where each factor is a linear polynomial with the same amount of factors as the degree of the leading term

Natural log

the inverse of e^x, base = natural number (e), the amount of time needed to reach a certain level of continuous growth

Frequency

the number of complete waves every 2 pi units, inversely related to period

sample space

the number of possible outcomes of an event

multiplicity

the number of values for which a given condition holds, specifically the number of times a given polynomial equation has a root at a given point

Conditional probability

the probability of an event ( A ), given that another ( B ) has already occurred

Partial fraction decomposition

the process of taking a rational expression and decomposing it into simpler rational expressions that we can add or subtract to get the original rational expression

Factorials

the product of a positive integer n and all positive integers below n

Inverse

the relation formed when the independent variable (x) is exchanged with the dependent variable (y) in a given relation, An inverse function [f(x)^-1] is the reflection of f(x) over the line y=x

Domain

the set of all first elements of ordered pairs (all possible x-values)

Range

the set of all second elements of ordered pairs (all possible y-values)

Intersection

the set of elements that belong to both sets A and B in an event

Collinear

three or more points that lie on a straight line

Isosceles

triangle with at least two equal sides (therefore at least two equal angles)

Scalene

triangles where every side is a different length

Rotation

turning around a center, measures in degrees or radians

Intercepts

where the function crosses the x and y axes, respectively

Pythagorean's theorem

"The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides.", the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

Indeterminate

(0/0), (of a quantity) having no definite or definable value, A value that cannot be precisely known or found out

Undefined

(any number/0), The x coordinate is constant no matter what y is, 'Undefined' does NOT have a value or its just not defined. 'Indeterminate' has a value which cannot be precisely known, Vertical

Rational root theorem

(factors of constant)/(factors of leading coefficient), helps find roots of a polynomial

Common Logarithm

A logarithm to the base of 10

Phase shift

A phase shift represents the amount a wave has shifted horizontally (or vertically) from the original wave

Vertex

A point where two or more straight lines meet

Fundamental theorem of algebra

A polynomial of degree n has n roots, every non-constant single-variable polynomial with complex coefficients has at least one complex root.

Ratio

A relationship or comparison between two numbers expressed as the number of times one is bigger or smaller than the other, Can be written using a colon read as "blank" to "blank", Can be written as a fraction or a decimal as well, Has no units, A proportion; for example, in the ratio 2:3, there could be two chocolate bars for every three marshmallows

Permutation

A selection of a group of objects in which order is important

Combinations

A way of selecting several things out of a group, where order does not matter

Geometric probability

Form of theoretical probability determined by a rati

Mutually Exclusive

Mutually exclusive is a statistical term used to describe a situation where the occurrence of one event is not influenced or caused by another event. In addition, it is impossible for mutually exclusive events to occur at the same time

Fraction

Part of a whole, Numerator- the top number over the dividing line that states how many parts there are, Denominator- the number below the dividing line that states how many equal parts there are from the whole

Theoretical probability

Ratio of (number of favorable outcomes) / (total number of outcomes)

Event

Set of outcomes

Asymptotes

Simply put: line that a curve approaches as but never touches it. An x or y value that the function can never be: Horizontal asymptote - a y value that the function can never have, Vertical asymptote - an x value that the function can never have

Game theory

a branch of mathematics that deals with the analysis of games (situations involving psttoe, In this context, what is the definition of a game?

Venn diagram

a diagram representing mathematical or logical sets pictorially as circles or closed curves within an enclosing rectangle (the universal set), common elements of the sets being represented by the areas of overlap among the circles.

Holes

a discontinuity in a graph (a singular point that the graph does not cover)

Composite functions

a function that consists of another function. f(g(x)) or fog

perpendicular

a line meets another line at a right angle (90º)

Axis of symmetry

a line that divides a graph so that either side look like mirror images of each other

Number sequence

a list of ordered numbers defined by a rule

Degrees

a measure of angles where an entire rotation is equal to 360º

Divisor

a number by which another number (dividend) is divided

Dividend

a number that is divided by another number (divisor)

Remainder theorem

a polynomial is completely divisible once by its factor to obtain a smaller polynomial and a remainder of zero, you can divide a polynomial by some linear factor (x - some number) to factorize and break down a polynomial of any degree

Average

Also called the arithmetic mean, it is the calculated central value of a set of numbers, To calculate: add up all the numbers and divide by the number of numbers there are

Amplitude

Amplitude is the height from the mean, or rest, value of the function to its maximum or minimum

Slope

Average rate of change

Fundamental counting principles

Determines the number of ways different events can occur. if one event has m possible outcomes, and a second independent event has n possible outcomes, then there are m*n outcomes together.

Outcome

Each possible result of an event, Equally likely outcomes - same chance of occurring

The Cartesian plane

It is a plane which has a horizontal line (x axis) and a vertical line (y axis). It is also known as coordinate plane, Created by Rene Descartes, who said "I think, therefore I am", Two real number lines intersecting at a right angles

Period

The distance required for the function to complete one full cycle, horizontal distance traveled before y-values begin to repeat

Quotient

The name of the resulting answer from a division problem

Piecewise function

function defined by multiple sub functions, each sub function applying to a certain interval of the main function's domain

Rational function

functions whose numerator and denominator are both polynomials, ratio of two polynomials (polynomial in denominator cannot equal zero)

Probability

how many times an outcome can occur compared to all possible outcomes, How likely an event is to occur

Angle bisector

line that divides an angle in half. context: you use an angle bisector segment when deriving the side lengths of a 30-60-90 triangle

Synthetic division

method of division by dividing by a linear factor

Direct versus inverse

must have a 1 to 1 relationship (every x-value has only 1 y-value and vice versa). can use the horizontal and vertical line tests to determine

improper rational expression

numerator is a higher or equal degree polynomial compared to the denominator

Favorable outcomes

outcomes in a specified event