Precalculus Definitions
Law of Sines
(sin A)/a = (sin B)/b = (sin C)/c
term
1. the monomials that make up a polynomial. 2. Each number in a sequence or series
combination
A arrangement of objects in which order is not important.
removable discontinuity
A characteristic of a function in which the function is continuous everywhere except for a hole at x=c.
continuity correction factor
A correction for continuity that must be used when approximating a binomial distribution.
class
A data value or group of data values
continuous function
A function that can be graphed with no breaks, holes, or gaps.
monomial function
A function that can be written as f(x) = a or f(x) = a*x^n, where a and n are nonzero constant real numbers.
radical function
A function that can be written as f(x) = nth root(x^p), where n and p are positive integers greater than 1 that have no common factors.
even function
A function that is symmetric with respect to the y-axis.
right Riemann sum
A method fro approximating the area under the curve by using the values at the right endpoints.
direct substitutions
A method of evaluating the limit of a polynomial or rational function f(x) as x approaches c by finding f(c)
triangle method
A method of finding the resultant vector by translating one vector so that its tail touches the tip of another
Cramer's Rule
A method that uses determinants to solve square systems of linear equations.
point estimate
A single value estimate of an unknown population parameter.
experiment
A situation involving chance or probability that leads to specific outcomes.
ordered triple
Coordinates of the location of a point in space given by real numbers (x, y, z).
bivariate data
Data with two variables
independent events
Events that do not affect each other
antiderivative
F(x) is an antiderivative of f(x)if F'(x) =f(x)
pth roots of unity
Finding the pth roots of 1.
root
For a function f(x), a solution of the equation f(x) = 0
minimum
For a function f, the least value of f(x). A critical point on the graph of a function where the curve changes from decreasing to increasing
ambiguous case
Given the measures of two sides and a nonincluded angle, either no triangle exists, exactly one triangle exists, or two triangles exist.
imaginary unit
I, or the principal square root of -1.
relative frequency
In a frequency table, the frequency of occurrence fro each data value.
alternative hypothesis
One of two hypotheses that need to be stated to test a claim; states that there is a difference between the sample value and the population parameter. The alternative hypothesis contains a statement of inequality as >, =, <
complement
The complement of an event A consists of all the outcomes in the sample space that are not included as the outcomes of event A.
polar form
The complex number z = a+bi written as z r*cisθ.
common difference
The difference between successive terms of an arithmetic sequence.
exponential series
The power series that approximates e^x.
confidence level
The probability that the interval estimate will include the actual population parameter.
factorial
The product of all positive integers that are less than or equal to n.
dependent polynomial
The quotient when a polynomial is divided by one of its binomial factors x-c.
class width
The range of values for each class of data.
common ratio
The ratio of successive terms of a geometric sequence.
equivalent vectors
Vectors that have the same magnitude and direction.
transformation
a change in the position or shape of the graph of a parent function.
unit circle
a circle of radius 1 centered at the origin of a coordinate system.
set
a collection of objects or numbers, often showing using braces {} and usually named by a capital letter.
scalar
a constant
normal distribution
a continuous probability distribution in which the graph of the curve is bell-shaped and symmetric with respect to the mean; the mean, median, and mode are equal and located at the center. The curve is continuous and approaches, but never touches, the x-axis; the total area under the curve is equal to 1 or 100%.
interval
a data value or group of data values
dimensions
a description of the number or rows and columns of a matrix.
t-distribution
a family of curves that are dependent on a parameter known as the degrees of freedom
quadratic function
a function of the form f(x) = a*x^2 + b*x + c, where a ≠ 0, with parent function f(x) = x^2
transcendental function
a function that cannot be expressed in terms of algebraic operations, such as an exponential or logarithmic function.
quartic function
a function that contains a fourth-degree polynomial
absolute value function
a function that contains an absolute value of the independent variable, with parent function f(x) = IxI
discontinuous function
a function that is not continuous
odd function
a function that is symmetric with respect with the respect to the origin.
logistic growth function
a function that models exponential growth with limiting factors. Logistic growth functions are bounded by horizontal asymptotes y = 0 and y = c, where c is the limit to growth.
midline
a horizontal axis that is the reference line about which the graph of a sinusoidal function oscillates
objective function
a linear function of the form f(x,y) = ax + by + c to be optimized in a two-dimensional linear programming problem.
invertible matrix
a matrix that has an inverse.
nonsingular matrix
a matrix that has an inverse.
sample
a part of a population
upper bound
a real number b that is greater than or equal to the greatest real zero of a polynomial function.
measures of spread
a representation of how spread out or scattered a set of data is.
translation
a rigid transformation that has the effect of shifting the graph of a function.
residual plot
a scatter plot of the residuals in which the horizontal line at zero corresponds to the regression line.
locus
a set of all point that fulfill a geometric property.
system of equations
a set of equations with the same variables
system of inequalities
a set of inequalities with the same variables.
synthetic division
a shortcut for dividing a polynomial by a linear factor of the form x-c.
resultant
a single vector that results when two or more vectors are added.
z-axis
a third axis in a three-dimensional coordinate system that passes through the origin and is perpendicular to both the x- and y-axes
unit vector
a vector that has a magnitude of 1 unit.
sequence
an ordered list of numbers
maximum
for a function f, the greatest value of f(x). A critical point on the graph of a function where the curve changes from increasing to decreasing.
clusters
subgroups of data
one-to-one
1. A function in which no x-value is matched wit more than one y-value and no y-value is matched with more than one x-value. 2. A function whose inverse is a function.
reciprocal function
1. A function of the form f(x) = 1/a(x), where a(x) is a linear function and a(x) ≠ 0, with parent function f(x) = 1/x 2. Trigonometric functions that are reciprocals of each other.
tangent
1. A line that intersects a circle at exactly one point. 2. In a right triangle with acute angle θ, the ratio comparing the length of the side opposite θ and the side adjacent to θ.
element
1. Each object or number in a set 2. Each entry in a matrix
standard position
1. In the coordinate plane, an angle positioned so that its vertex is at the origin and its initial side is along the positive x-axis. 2. A vector that has its initial point at the origin .
parameter
1. arbitrary values, usually time or angle measurement, used in parametric equations. 2. A measure that describes a characteristic of a population.
vertex
1. the common endpoint of two or more noncollinear rays. 2 A point at which a parabola intersects its axis of symmetry. 3. The two endpoints of the major axis of a ellipse.
infinite discontinuity
A characteristic of a function in which the absolute value of the function increases or decreases indefinitely as x-values approach c from the left and right.
jump discontinuity
A characteristic of a function in which the function has two distinct limit values as x-values approach c from the left an right.
standard form
A complex number written in the form a+bi.
absolute value of a complex number
A complex number's distance from zero in the complex plane
three-dimensional coordinate system
A coordinate system formed by three perpendicular number lines, the x-, y-, and z-axes. that intersects at the origin O. Each point is represented by an ordered triple of real numbers (x,y,z).
polar coordinate system
A coordinate system in which the location of a point is identified by polar coordinates of the form (r,θ), where r is the distance from the center, or the pole, to the given point and θ is the measure of the angel formed by the polar axis and a line from the pole through the point.
quadrant bearing
A directional measurement of a vector between 0 and 90 east or west of the north-south line.
true bearing
A directional measurement of a vector where the angle is measured clockwise from north.
binomial probability distribution function
A discrete function of the random variable X, represented in the binomial probability formula.
sampling distribution
A distribution of the means of random samples of a certain size that are taken from a population.
conic section
A figure that is formed when a plane intersects a double-napped right cone, also called a conic.
rational function
A function of the form f(x) = a(x)/b(x, where a(x) and b(x) are polynomial functions, and b(x) ≠ 0.
exponential function
A function of the form f(x) = a*b^x, where x is any real number and a and b are real number constants such that a ≠ 0, b is positive, b ≠ 1.
polynomial function
A function of the form f(x) = a*x^n + b*x^(n-1) ..., where a,b,c ... are real numbers.
power function
A function of the form f(x) = a*x^n, where a and n are nonzero real numbers.
cubic functions
A function of the form f(x) = ax^3 + bx^2 + cx + d, where a =not 0, with parent function f(x) = x^3.
constant function
A function of the form f(x) = c, where c is any real number.
logarithm function with base b
A function of the form y = log (b,x), where b > 0, b ≠1, and x > 0, which is the inverse of the exponential function of the form b^y = x.
square root function
A function that contains a square root of the independent variable, with parent function f(x) = sqrt(x)
piecewise-defined function
A function that is defined using two or more expressions for different intervals of the domain.
algebraic function
A function with values that are obtained by adding, subtracting, multiplying, or dividing constants and the independent variable to a rational power
periodic function
A function with values that repeat at regular intervals. There exists a positive real number c such that f(t+c) = f(t) for all values t in the domain of f.
Bimodal Distribution
A graph of a distribution of data that has two modes
axis of symmetry
A line about which a figure is symmetric. In a parabola, the axis of symmetry is perpendicular to the directrix and passes through the focus.
line of best fit
A line drawn through a set of data points that describes how the response variable y changes as the explanatory variable x changes. Also called a regression line.
regression line
A line drawn through a set of data points that describes how the response variable y changes as the explanatory variable x changes. Also called the line of best fit.
asymptote
A line or curve that a graph approaches.
common logarithm
A logarithm with base 10, usually written log x
natural logarithm
A logarithm with base e, written ln x.
zero matrix
A matrix in which every element is zero.
row-echelon form
A matrix is in row-echelon form if the following conditions are met. 1. Rows of all zeros appear at the bottom of the matrix 2. The first nonzero entry in any row is 1. 3. For two successive rows with nonzero entries, the leading 1in the higher row is farther to the left than the leading 1 in the lower row.
coefficients matrix
A matrix that contains only the coefficients of a system of linear equations
augmented matrix
A matrix that contains the coefficients and constant terms of a system of linear equation, written in standard form with the constant terms to the right of the equal sign.
singular matrix
A matrix that does not have a inverse.
column matrix
A matrix that has only one column.
square matrix
A matrix with the same number of rows an columns.
eccentricity
A measure that determines how "circular" or "stretched" an ellipse will be.
sample correlation coefficient
A measure that determines the type ad strength of the linear relationship between the variables in bivariate data that represent a sample of the population.
correlation coefficient
A measure that determines the type and strength of the linear relationship between the variables in bivariate data.
standard normal distribution
A normal distribution of z-values with a mean of 0 and a standard deviation of 1.
sigma notation
A notation in order to expresses a sum of numbers and sequences.
step function
A piecewise-defined function in which the graph is a series of line segments that resemble a set of stairs.
complex plane
A plane used to graph complex numbers. The real component is graphed on the horizontal axis and the imaginary part is graphed on the imaginary axis.
degenerate conic
A point, a line, or two intersecting lines that are formed when a plane intersects the vertex of a double-napped right cone.
quadratic equation
A polynomial equation of degree two, in the form a*x^2 + b*x + c, where a ≠ 0.
quadratic form
A polynomial expression that is written in the form a*u^2 + b*u + c fro any real number a, b, and c, where a ≠ 0 and u is some expression in x.
Fundamental Theorem of Algebra
A polynomial function of degree n, where n > 0 , has at least one zero(real or imaginary) in the complex number system.
binomial experiment
A probability experiment in which there are a fixed number of independent trials, there are exactly two possible outcomes for each trial, and the probability of success is the same for each trial
completing the square
A process used to make a quadratic expression into a perfect square trinomial.
vector
A quantity that has both magnitude and direction.
interval estimate
A range of values used to estimate an unknown popular parameter
partial fraction decomposition
A rational expression rewritten as the sum of two simpler rational functions.
lower bound
A real number a that is less than or equal to the least real zero of a polynomial function.
unbounded
A region formed by a system of linear inequalities in a linear programming problem.
function
A relation that assigns to each element in the domain exactly one element in the range.
Descartes' Rule of Signs
A rule that gives information about the number of positive and negative real zeros of a polynomial function by looking at a polynomial's variations in sign.
inferential statistics
A sample of data is analyzed and conclusions are made abut the entire population.
arithmetic sequence
A sequence in which the difference between successive terms is a constant.
Fibbonacci Sequence
A sequence in which the first two terms are 1 and each of the additional terms is the sum of the two previous terms.
geometric sequence
A sequence in which the ratio between successive terms is a constant.
finite sequence
A sequence that as a finite number of terms
infinite sequence
A sequence that has infinitely many terms.
empty set
A set with no elements
extraneous solution
A solution that does not satisfy the original equations.
confidence interval
A specific interval estimate of a parameter in an experiment that can be found when the maximum error of estimate is added to and subtracted from the sample mean
directrix
A specific line from which all points on a parabola are equidistant.
five-number summary
A statistic that includes the minimum value, lower quartile, median, upper quartile, and the maximum vale of a data set.
resistant statistic
A statistic that is not highly affected by the presence of outlying data values.
consistent
A system of equations that has at least one solution
inconsistent
A system of equations that has no real solutions.
multivariable linear system
A system of linear equations in two or more variables, also called a multivariable linear system.
square system
A system of linear equations that has the same number of equations as variables.
frequency distribution
A table used to organize data by group, classes, or intervals.
probability distribution
A table, equation, or graph that links each possible value for a random variable with its probability of occurring.
reflection
A transformation in which a mirror image of the graph of a function is produced with respect to a specific line.
dilation
A transformation in which the graph of a function is compressed or expanded vertically or horizontally.
oblique triangle
A triangle that is not a right triangle
Pascal's Triangle
A triangular array of number such that the first and last numbers in each row are 1 and every other number is formed by adding the two number immediately above that number in the previous row. The (n+1)the row contains the coefficients of the terms of the expansion (a+b)^n for natural number n.
circular function
A trigonometric function defined as a function of the real number system using the unit circle.
radians
A unit of angular measurement equal to 180/π or 57.296.
continuous random variable
A variable that can take on an infinite number of possible values within a specified interval in a probability experiment
torque
A vector quantity that measures how effectively a force applied to a lever causes rotation along the axis of rotation.
component form
A vector represented by its rectangular components.
damped wave
A wave whose amplitude decreases, such as the graph of a damped trigonometric function.
dot product
Also known as a scalar product, it finds a number value from two vectors.
quadrantal angle
An angle in standard position that has a terminal side that lies on one of the coordinate axis.
correlation
An area of inferential statistics that involves determining whether a relationship exists between two variables.
permutation
An arrangement of objects in which order is important.
population
An entire group of living things or objects.
polar equation
An eqation expressed in terms of polar coordinates
function notation
An equation of y in terms of x can be rewritten so that y = f(x). For example, y = 4x can be written as f(x) = 4x.
trigonometric identity
An equation that involves trigonometric functions that is true for all values of the variables.`
indeterminate form
An expression obtained when evaluating a limit that does not give enough information to determine the original limit.
set-builder notation
An expression that describes a set of numbers by using the properties of numbers in the set to define the set, for example { x I x > 8 , x e W}
reduction identity
An identity that results when a sum or difference identity is used to rewrite a trigonometric expression in which one of the angles is a multiple of 90 degrees or π/2 radians.
pure imaginary number
An imaginary number (a+bi), where a=0
polynomial inequality
An inequality comparing f(x) and 0, where f(x) is a polynomial function.
rational inequality
An inequality that contains one or more rational expressions
definite integral
An integral that has an upper and lower bound.
coterminal angles
Angles in standard position that have the same initial and terminal sides, but different measures.
complex number
Any number that can be written in the form a+bi where a and b are real numbers and I is the imaginary part.
sinusoid
Any transformation of a sine function.
hypothesis test
Assesses evidence provided by data about a claim concerning a population parameter.
decreasing
Describes a function f or an interval of a function in which for any two points, a positive change in x results in a negative change in f(x).
increasing
Describes a function f or an interval of a function in which for any two points, a positive change in x results in a positive change in f(x).
empirical rule
Describes areas under the normal curve over intervals that are one, two, and three standard deviations from either side of the mean,
point symmetry
Describes graph that can be rotated 180 degrees with respect to a point and appear unchanged.
Rational Zero Theorem
Describes how the leading coefficients and constant terms of a polynomial function with integer coefficients can be used to determine a list of possible rational zeros.
polar coordinate
Describes the location of a point P(r,θ) in the polar coordinate system, where r is the directed distance from the pole O to the point and θ is the directed angle from the polar axis OP.
end behavior
Describes what happens to the value of f(x) as x increases or decreases without bound
second differences
Differences that are found by subtracting consecutive first differences from one another
percentiles
Divide a distribution into 100 equal groups and are symbolized by P1, P2, ... The nth percentile or Pn, is the vale such that n% of the data are lower than Pn.
period
For a function y = f(t), the smallest positive number c for which f(t+c) = (t).
phase shift
For a sinusoidal function , the difference between the horizontal position of a function and that of an otherwise similar sinusoidal function.
vertical shift
For a sinusoidal function, a vertical translation that is the average of the maximum and minimum values of the function.
frequency
For a sinusoidal function, the number of cycles the function completes in a one unit interval. The frequency is the reciprocal of the period.
Euler's Formula
For any real number θ, e^iθ = cisθ.
amplitude
Half the distance between the maximum and minimum values of a sinusoidal function.
greatest integer function
Has the parent function f(x) = {x}, which is defined as the greatest integer less than or equal to x.
multiplicity
If (x-c)^m is the highest power of (x-c) that is a factor of polynomial function f, then c is a zero of multiplicity m of f , where m is a natural number.
work
If a constant force F acts on an object to move it from point A to point B, then the work done equals the dot product of the constant force F and the directed distance AB or F dot AB.
diverge
If a sequence does not have a limit, it is said to diverge.
converge
If a sequence has a limit such that the terms approach a unique number, it is said to converge.
subset
If every element of set B a set is contained in set A, then B is a subset of A.
parametric curve
If f and g are continuous functions, then the set of ordered pairs (f(t),g(t)) is a plane curve with x = f(t) and y = g(t) as the parametric equations and t as the parameter.
Fundamental Theorem of Calculus
If f is continuous on [a,b] and F(x) is any antiderivative of f(x), then the integral from a to b of f(x) = F(b) - F(a).
Heron's formula
If triangle ABC has side lengths a,b, and c, then the area of the triangle can be found
inverse cosine
If θ is an acute angle and cos θ = x, then the inverse cosine of x, or cos^-1 x,is the measure of angle θ.
inverse sine
If θ is an acute angle and sin θ = x, then the inverse sine of x, or sin^-1 x, is the measure of angle θ.
inverse tangent
If θ is an acute angle and tan θ = x, then the inverse tangent of x, or tan^-1 x, is the measure of angle θ.
sector
In a circle the region bounded by a central angle its intercepted arc.
damping factor
In a damped trigonometric function of the form y = f(x) sin(bx) or y = f(x) cos(bx), f(x) is the damping factor.
symmetrical distribution
In a data distribution, the data are evenly distributed on both sides of the mean.
positively skewed distribution
In a data distribution, the mean is greater than the median, the majority of the data are on the left, and the tail extends to the right.
negatively skewed distribution
In a data distribution, the mean is les than the median, the majority of the data are on the right, and the rail extends to the left.
implied domain
In a function with an unspecified domain, the set of all real numbers for which the expression used to define the function is real.
relevant domain
In a function, the part of the domain that is relevant to a model.
dependent variable
In a function, the variable, usually y, that represents any value in the range.
independent variable
In a function, the variable, usually' x, that represents any value in the domain.
leading coefficient
In a polynomial function, the coefficient of the variable with the greatest exponent.
secant
In a right triangle with acute angle θ, the ratio comparing length of the hypotenuse to the side adjacent to θ. It is the reciprocal of the cosine ratio, or sec θ = 1/cos θ.
cosecant
In a right triangle with acute angle θ, the ratio comparing the length of the hypotenuse to the side opposite of θ. It is the reciprocal of the sine ratio, or csc θ = 1/sinθ.
cosine
In a right triangle with acute angle θ, the ratio comparing the length of the side adjacent to θ and the hypotenuse.
cotangent
In a right triangle with acute angle θ, the ratio comparing the length of the side adjacent to θ and the side opposite θ. It is the reciprocal of the tangent ratio, or cotθ = 1/tanθ
sine
In a right triangle with acute angle θ, the ratio comparing the length of the side opposite θ and the hypotenuse.
real part
In an imaginary number a + bi, a is the real part.
imaginary part
In an imaginary number a+bi, b is the imaginary part.
inductive step
In mathematical Induction, showing that something works for the case after Pk, or showing that P(k+1) is true.
inductive hypothesis
In mathematical induction, assuming that something works for any particular case, or that assuming that Pk is true
anchor step
In mathematical induction, showing that something works for the first case, or that P1 is true
regular partition
In the area under the graph of a function, an interval that is subdivided into equal subintervals.
logarithm
In the function x = b^y, y is called the logarithm, base b, of x. Usually written y = log(b,x) and is read log base b of x.
extended principle of mathematical induction
Instead of verifying that Pn is true for n = 1, as in principle of mathematical induction, instead verify Pn is true for the first possible case.
inverse
Let A be an n*n matrix. If there exists a matrix B such that AB = BA = In, then B is called the inverse of A and is written as A^-1
principle of mathematical induction
Let Pn be a statement about a positive integer n. Then Pn is true for all positive integers n is and only if: 1. P1 is true, and 2. for every positive integer k, is Pk is true, then P(k+1) is true
vector projection
Let u and v be nonzero vectors, and let w1 and w2 be vector components of u such that w1 is parallel to v. Then vector w1 is called the vector projection of u onto v denoted proj(v,u) and proj(v,u) = ((u .v)/(IvI^2))v
sampling error
Occurs when a sample is not a complete representation of the population and causes differences between sample means and the population mean.
holes
Removable discontinuities on the graph of a function that occur when the numerator and denominator of the function have common factors. The holes occur at the zeros of the common factors.
random variable
Represents a numerical value assigned to an outcome of a probability experiment.
Guass - Jordan Elimination
Solving a system of linear equations by transforming an augmented matrix so that it is in reduced row-echelon form.
angle of elevation
The angle formed by a horizontal line and an observer's line of sight to an object above.
angle of depression
The angle formed by a horizontal line and an observer's line of sight to an object below.
argument
The angle θ of a complex number written in the form r(cisθ)
Binomial coefficients
The coefficients of the terms of an expanded binomial (a+b)^n
composition
The combining of functions by using the result of one function to evaluate a second function. f(g(x))
derivative
The derivative of the function f(x) is the function f '(x) and defines the rate of change of the tangent line at a given point.
range
The difference between the greatest and least values in a set of data.
Binomial Distribution
The distribution of the outcomes of a binomial experiment and their corresponding probabilities
co-vertices
The endpoints of the minor axis of an ellipse.
identity function
The function f(x) = x, which passes through all points with coordinates (a,a).
cardioid
The graph of a polar equation of the form r = a +- a*cosθ, where a is a positive integer.
limacon
The graph of a polar equation of the form r = a +- bcosθ or r = a +- bsinθ, where a and b are both positive.
rose
The graph of a polar equation of the form r = a*cos nθ or r = a*sin nθ, where n => is an integer
spiral of Archimedes
The graph of a polar equation of the form r = aθ + b
lemniscates
The graph of a polar equation of the form r^2 = a^2*cos2θ or r^2 = a^2*sin2θ.
right-tailed test
The hypothesis test if Ha: μ > κ
indefinite integral
The indefinite integral of f(x) is defined by int(f(x)) = F(x) +C
explanatory variable
The independent variable x in bivariate data.
arccosine function
The inverse function of cosine that has a domain of [-1,1] and a range of [0,pi]
arcsine function
The inverse function of sine that has a domain of [-1,1] and a range of [-pi/2,pi/2]
arctangent function
The inverse function of tangent that has a domain of [-1,1] and a range of [0,pi]
horizontal asymptote
The line y=c is a horizontal asymptote of the graph of f if its limit is c as x increases or decreases without bound.
extrema
The maximum and minimum value of a function.
expected value
The mean of the random variable in a probability distribution.
center of an ellipse
The midpoint of the major and minor axes of an ellipse.
elementary row operations
The operations shown below are used to transform an augmented matrix into an equivalent matrix. 1. Interchange any two rows 2. Multiply one row by a nonzero real number. 3. Add a multiple of one row to another row
conjugate axis
The segment that is perpendicular to the transverse axis of a hyperbola, passes through the center, and has a length of 2b units.
average rate of change
The slope of the line through any two points on the graph of a nonlinear function f.
Quadratic Formula
The solutions of a quadratic equation of the form a*x^2 + b*x + c, where a ≠ 0, are given by the Quadratic Formula.
standard error of the mean
The standard deviation of the sample means, given by σx = σ/sqrt(n)
cumulative frequency
The sum of a frequency and all frequencies of previous classes.
linear combination
The sum of two vectors, each multiplied by a scalar, that is used to represent a vector with a given initial point and terminal point.
tangent line
The tangent line to f(x) at x is the line passing through the point (x,f(x)) with slope m, where m is the derivative of f(x).
arithmetic means
The terms between two nonconsecutive terms of an arithmetic sequence.
imaginary axis
The vertical axis of a complex plane on which the imaginary component of a complex number system.
critical values
The z-values that correspond to a particular confidence level.
extrapolation
To use the equation of the least-squares regression line to make the predictions far outside the range of the x-values that were used to obtain the regression line.
linearize
Transform data so that they appear to cluster about a line by applying a function to one or both of the variables in the data set.
complex conjugates
Two complex number of the form a+bi and a-bi where b is not equal to zero.
foci
Two fixed points used to define an ellipse or hyperbola.
inverse function
Two function f and f^-1 are inverse functions if and only if f[f^-1(x)] = x fro every x in the domain of f^-1(x), and f^-1[f(x)] = x fro every x in the domain of f(x).
components
Two or more vectors with a sum that is a given vector.
inverse relation
Two relations are inverse relations if and only if one relation contains the element (b,a) whenever the other relation contains the element (a,b).
leading term test
Uses the power and coefficients of the leading term of a polynomial to determine the end behavior of a polynomial function.
percentile graph
Uses the same vales as a cumulative relative frequency graph, except that the proportions are instead expressed as percents.
Gaussian elimination
Using the operations below to transform a system of linear equations into an equivalent system. 1. Interchange any two equations 2. Multiply on of the equations by a nonzero real number 3. Add a multiple of one equation to another equation.
first differences
Values obtained by subtracting each term in a sequence from its successive term.
opposite vectors
Vectors that have the same magnitude but opposite direction.
Conjugate Root Theorem
When a polynomial equation in one variable has a zero of the form a+bi, then its complex conjugate, a-bi is also a root.
independent
When a system of linear equations has exactly one solution.
recursive formula
a formula used to determine the nth term of a sequence using one or more of the preceding terms.
row matrix
a matrix that has only one row.
parallelogram method
a method of finding the resultant vector by translating one vector so that its tail touches the tail of another. A parallelogram is drawn and the diagonal is the resultant vector.
substitution method
a method of solving a system of equations in which one equation is solved for one variable in terms of the other.
measure of central tendency
a number that represents the center or middle of a data set.
turning point
a point on the graph of a function that indicates where the graph changes from increasing to decreasing, or vice versa. The location of a relative maximum or minimum
parallelepiped
a polyhedron with faces that are all parallelograms.
irreducible over the reals
a quadratic expression that has real coefficients but no real zeros associated with it.
discrete random variable
a random variable that can take on a finite number of possible values in a probability experiment.
Law of Cosines
c^2 = a^2 + b^2 - 2ab*cosC
constraints
conditions given to variables in a two-dimensional linear programming problem, often expressed as a system of linear inequalities
outliers
data that are more than 1.5 times the interquartile range beyond the upper and lower quartiles
univariate data
data with one variable
constant
describes a function s or an interval of a function in which for any two points, a positive change in x results in a zero change in f(x)
line symmetry
describes graphs that can be folded along a line so that two halves match exactly.
nonremovable discontinuity
describes infinite and jump discontinuities because they cannot be eliminated by redefining the function at that point.
octants
eight regions into which the three axes of a three-dimensional coordinate system divide space.
elimination method
eliminate one of the variables in a system of equations by adding or subtracting the equations.
nth root
for any real number a and b, and any positive integer n, if a^n = b, then a is the nth root of b.
instantaneous rate of change
for the graph of f(x), the slope m of the line tangent at the point (x,f(x)) given by the f '(x), provided that it exists
rectangular components
horizontal and vertical components of a vector.
Linear Factorization Theorem
if f(x) is a polynomial function of degree n > 0, then f has exactly n linear factors and f(x) = a(x-c)(x-c) .., where a is some nonzero real number and c's are the complex zeros of f.
continuous compound interest
interest that is reinvested continuously so that there is no waiting period between interest payments.
null hypothesis
one of two hypotheses that need to be stated to test a claim; states that there is not a significant difference between the sample value and the population parameter. The null hypothesis contains a statement of equality such as >=, =, or <=.
orientation
plotting points of parametric equations in the order of increasing values of t traces the curve in a specific direction of the curve.
trigonometric ratios
ratios that are formed using the side measures of a right triangle and a reference angle θ.
degrees of freedom
represent the number of values that are free to vary after a sample statistic is determined, and are equal to n-1 in a sample of n values.
z-value
represents the number of standard deviations that a given data value is from the mean. Also known as the z-score and z-test statistic
multiple optimal solutions
solutions that occur when the graph of the equation related to the objective function f to be optimized is coincident with one side of the region of feasible solutions.
triple scalar product
t-(u*v) the scalar product of a vector t and the cross product of vectors u and v.
modulus
the absolute value of a complex number, the number r when a complex number is written in the form r*cisθ.
reference angle
the acute angle formed by the terminal side of an angle in standard position and the x-axis.
standard deviation
the average amount by which individual items deviate from the mea of all the data found by taking the square root of the variance and represented by σ.
Argand Plane
the complex plane
response variable
the dependent variable in bivariate data.
residual
the difference between an observed y-value of a data point and its predicted y-value on a regression.
determinant
the difference of the product f the two diagonals of the matrix.
direction
the directed angle between the vector and the horizontal line that could be used to represent the positive x-axis.
terminal point
the ending point of a vector that is represented by a directed line segment. Also known as the head or tip of the vector
vertices
the endpoints of the major axis of an ellipse.
terminal side
the final position of a ray after rotation when forming an angle
damped trigonometric function
the function formed when a sinusoidal function y = sin(bx) or y = cos(bx) is multiplied by another function y = f(x). A function of the form y = f(x) sin(bx) or y = f(x) cos(bx)
zero function
the function sometimes known as the zero function is the constant function with constant c = 0. In other words, f(x) = 0.
real axis
the horizontal axis of a complex plane on which the real component on a complex number is graphed.
left-tailed test
the hypothesis test if Ha: μ < κ
two-tailed t-test
the hypothesis test is Ha: μ ≠ κ
identity matrix
the identity matric of order n, In is an n*n matrix consisting of all 1s on its main diagonal, from upper left to lower right, and 0s for all other elements.
intersection
the intersection of sets A and B is all elements found in both A and B
inverse trigonometric function
the inverse sine of x or sin-1 x, the inverse cosine of x or cos^-1n x, and the inverse tangent of x or tan^-1 x.
natural base
the irrational number e, which is approximately equal to 2.718281828...
magnitude
the length of the directed line segment that represents the vector.
one-sided limit
the limit L1 of f(x) as x approaches c from the left or the limit L2 of f(x) as x approaches c from the right.
two-sided limit
the limit of f(x) as x approaches c from the left and from the right, which exists only when both one-sided limits exist and are equal.
least-squares regression line
the line for which the sum of the squares of the residuals is at a minimum.
latus rectum
the line segment that passes through the focus of a parabola, is perpendicular to the axis of symmetry, and has endpoints on the parabola
secant line
the line through two points on a curve.
vertical asymptote
the line x = c is a vertical asymptote f the graph of f if the limit as x approaches c is infinitely large.
ellipse
the locus of points in a plane such that the sum of the distances from two fixed points, called foci, is constant.
parabola
the locus of points in a plane that are equidistant from a fixed point, called the focus, and a specific line, called the diretrix.
lower limit
the lower bound of a definite integral.
p-value
the lowest level of significance at which H0 can be rejected for a given data set.
level of significance
the maximum allowable probability of committing a Type I error, denoted α
maximum error of estimate
the maximum difference between the point estimate and the actual value of the parameter in an experiment.
variance
the mean of the squares of the deviations from the arithmetic mean.
median
the middle number in a set if data when the data are arranged in numerical order. If the data set has an even number, the median is the mean of the two middle numbers.
damped harmonic motion
the motion of an objet whose amplitude decreases with time due to friction.
inverse matrix
the multiplicative inverse of a square matrix. The product of a matrix A and its inverse A^-1 must equal the identity matrix In.
principal nth root
the nonnegative nth root.
mode
the number(s) that appears most often in a set of data.
pole
the origin of the polar coordinate system, O.
integration
the process of evaluating an intagral
optimization
the process of finding a minimum or maximum value for a specific quantity., usually to minimize costs in order to maximize profits in business.
linear programming
the process of finding a minimum or maximum value of a linear function for a religion defined by linear inequalities.
differentiation
the process of finding the derivative of a function.
interquartile range
the range of the middle half of a set of data. It is the difference between the upper quartile and the lower quartile.
linear speed
the rate at which an object moves along a circular path.
angular speed
the rate at which the object rotates about a fixed point
cumulative relative frequencies
the ratio of the cumulative frequency of the class to all the data.
damped oscillation
the reduction in amplitude of a sinusoidal wave of a damped trigonometric function.
repeated zero
the related zero c of a function when a factor (x-c) occurs more than once in the completely factored form of f(x).
differential equation
the result when finding the derivative of a function.
zero vector
the resultant when two opposite vectors are added, has a magnitude of 0 and no specific direction. Also called the null vector, denoted by 0.
statistics
the science of collecting, analyzing, interpreting, and presenting data.
major axis
the segment that contains the foci of an ellipse and has endpoints on the ellipse.
transverse axis
the segment that has a length of 2a units and connects the vertices of a hyperbola.
minor axis
the segment through the center of an ellipse that is perpendicular to the major axis and has endpoints on the ellipse
hyperbola
the set of all points in a plane such hat the absolute value of the difference of the distances from two foci is constant.
polar graph
the set of all points with coordinates (r,θ) that satisfy a given polar equation.
universal set
the set of all possible elements for a situation.
sample space
the set of all possible outcomes of an experiment.
feasible solutions
the set of possible solutions of a system of inequalities in a linear programming problem, which are points of the form (x,y).
parent function
the simplest function in a family of functions. A function that is transformed to create other members in a family of functions.
initial point
the starting point of a vector that is represented by a directed line segment, Also known as the tail of a vector.
initial side
the starting position of a ray when forming an angle
series
the sum of all the terms of a finite or infinite sequence.
mean
the sum of numbers in a set of data divided by the number of items in the data set.
finite series
the sum of the first n terms of a finite or infinite sequence.
nth partial sum
the sum of the fist n terms of a finite or infinite sum.
geometric series
the sum of the terms of a geometric sequence.
arithmetic series
the sum of the terms of an arithmetic sequence
infinite series
the sum of the terms of an infinite sequence.
geometric means
the terms between two nonconsecutive terms of a geometric sequence.
union
the union of sets A and B is all elements in both A and B, written as A U B.
limit
the unique value that a function approaches as x-values of the function approach c from the left and right sides.
upper limit
the upper bound of a definite integral
synthetic substitution
the use of synthetic division to evaluate a function
quartiles
the values that divide a set of data into four equal parts.
instantaneous velocity
the velocity achieved at a specific point in time.
cross products
the way to find the vector perpendicular to the plane containing the two vectors
zeros
the x-interceps of the graph of a function
solve a right triangle
to find the measures of all of the sides and angles of a right triangle.
verify an identity
to prove that both sides of the equation are equal for all values of the variables for which both sides are defined
interpolation
to use the equation of the least-squares regression line to make predictions over the range of the data.
equal matrices
two matrices that have the same dimensions and each element of one matrix is equal to the corresponding element of the other matrix.
dependent events
two or more events in which the outcome of one event affects the outcome of the other events.
orthogonal
two vectors with a dot product 0.
sign chart
used to determine on which intervals a polynomials function is positive or negative.
parallel vectors
vectors that have the same or opposite direction, but not necessarily the same magnitude.
partial fraction
when a rational function is written as the sum of two fractions with denominators that are linear factors of the original denominator, each fraction in the sum is a partial fraction
dependent
when a system of linear equations has an infinite number of solutions.
oblique asymptote
an asymptote that is neither horizontal nor vertical that occurs when the degree of the numerator of a rational function is exactly one more than the degree of the denominator. Also called a slant asymptote.
reduced row-echelon form
an augmented matrix in which the first nonzero element of each row of the coefficient portion od the matrix is 1 and the rest of the elements in the same column as this element are 0.
identity
an equation in which the left side is equal to the right side for all values of the variables for which both sides are defined.
parametric equation
an equation that can express the position of an object as a function of time.
interval notation
an expression that uses inequalities to describe subsets of real numbers
influential
an individual data point that substantially changes a regression.
power series
an infinite series of the form a0 + a1*x + a2*x^2 + ... where a and an can take on an values for n = 0,1,2,...
polar axis
an initial ray from the pole in the polar coordinate system, usually horizontal and directed toward the right.
differential operator
an operator such as d/dx, which specifies the action of taking the derivative of a function.
matrix
any rectangular array of variables or constants in horizontal rows and columns
