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
