an equation in which the variable appears in absolute value bars, typically with two solutions, one accounting for the positive expression and one for the negative expression
addition method
An algebraic technique used to solve systems of linear equations in which the equations are added in a way that eliminates one variable, allowing the resulting equation to be solved for the remaining variable; substitution is then used to solve for the first variable
addition principle
if one event can occur in [latex]m[/latex] ways and a second event with no common outcomes can occur in [latex]n[/latex] ways, then the first or second event can occur in [latex]m+n[/latex] ways
algebraic expression
constants and variables combined using addition, subtraction, multiplication, and division
annual percentage rate (APR)
the yearly interest rate earned by an investment account, also called nominal rate
annuity
an investment in which the purchaser makes a sequence of periodic, equal payments
area
in square units, the area formula used in this section is used to find the area of any two-dimensional rectangular region: [latex]A=LW[/latex]
arithmetic sequence
a sequence in which the difference between any two consecutive terms is a constant
arithmetic series
the sum of the terms in an arithmetic sequence
arrow notation
a way to symbolically represent the local and end behavior of a function by using arrows to indicate that an input or output approaches a value
associative property of addition
the sum of three numbers may be grouped differently without affecting the result; in symbols, [latex]a+\left(b+c\right)=\left(a+b\right)+c[/latex]
associative property of multiplication
the product of three numbers may be grouped differently without affecting the result; in symbols, [latex]a\cdot \left(b\cdot c\right)=\left(a\cdot b\right)\cdot c[/latex]
augmented matrix
a coefficient matrix adjoined with the constant column separated by a vertical line within the matrix brackets
average rate of change
the difference in the output values of a function found for two values of the input divided by the difference between the inputs
axis of symmetry
a vertical line drawn through the vertex of a parabola around which the parabola is symmetric; it is defined by [latex]x=-\frac{b}{2a}[/latex].
base
in exponential notation, [latex]a^n[/latex] the expression that is being multiplied
binomial
a polynomial containing two terms
binomial coefficient
the number of ways to choose [latex]r[/latex] objects from [latex]n[/latex] objects where order does not matter; equivalent to [latex]C\left(n,r\right)[/latex], denoted [latex]\left(\begin{gathered}n\\ r\end{gathered}\right)[/latex]
binomial expansion
the result of expanding [latex]{\left(x+y\right)}^{n}[/latex] by multiplying
binomial theorem
a formula that can be used to expand any binomial
break-even point
The point at which a cost function intersects a revenue function; where profit is zero
carrying capacity
in a logistic model, the limiting value of the output
Cartesian coordinate system
a grid system designed with perpendicular axes invented by René Descartes
center of an Ellipse
The midpoint of both the major and minor axes
center of a Hyperbola
The midpoint of both the transverse and conjugate axes of a hyperbola
change-of-base formula
a formula for converting a logarithm with any base to a quotient of logarithms with any other base
coefficient
any real number [latex]{a}_{i}[/latex] in a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
coefficient matrix
a matrix that contains only the coefficients from a system of equations
column
a set of numbers aligned vertically in a matrix
combination
a selection of objects in which order does not matter
common difference
the difference between any two consecutive terms in an arithmetic sequence
common logarithm
the exponent to which [latex]10[/latex] must be raised to get [latex]x[/latex]; [latex]{\mathrm{log}}_{10}\left(x\right)[/latex] is written simply as [latex]\mathrm{log}\left(x\right)[/latex]
common ratio
the ratio between any two consecutive terms in a geometric sequence
commutative property of addition
two numbers may be added in either order without affecting the result; in symbols, [latex]a+b=b+a[/latex]
commutative property of multiplication
two numbers may be multiplied in any order without affecting the result; in symbols, [latex]a\cdot b=b\cdot a[/latex]
complement of an event
the set of outcomes in the sample space that are not in the event [latex]E[/latex]
completing the square
a process for solving quadratic equations in which terms are added to or subtracted from both sides of the equation in order to make one side a perfect square
complex conjugate
the complex number in which the sign of the imaginary part is changed and the real part of the number is left unchanged; when added to or multiplied by the original complex number, the result is a real number
complex number
the sum of a real number and an imaginary number, written in the standard form [latex]a+bi[/latex], where [latex]a[/latex] is the real part, and [latex]bi[/latex] is the imaginary part
complex plane
a coordinate system in which the horizontal axis is used to represent the real part of a complex number and the vertical axis is used to represent the imaginary part of a complex number
complex roots
Solutions to a quadratic equation that contain imaginary numbers, occurring when the discriminant is negative
composite function
the new function formed by function composition, when the output of one function is used as the input of another
compound inequality
a problem or a statement that includes two inequalities
compound interest
interest earned on the total balance, not just the principal
conic Section
Any shape resulting from the intersection of a right circular cone with a plane
conjugate Axis
The axis of a hyperbola that is perpendicular to the transverse axis and has the co-vertices as its endpoints
conjugate pair
Two complex numbers in the form [latex]a + bi[/latex] and [latex]a - bi[/latex] that occur as solutions to quadratic equations with complex roots
consistent system
A system for which there is a single solution to all equations in the system and it is an independent system, or if there are an infinite number of solutions and it is a dependent system
constant
a quantity that does not change value
constant of variation
the non-zero value [latex]k[/latex] that helps define the relationship between variables in direct or inverse variation
continuous function
a function whose graph can be drawn without lifting the pen from the paper because there are no breaks in the graph
cost function
The function used to calculate the costs of doing business; it usually has two parts, fixed costs and variable costs
correlation coefficient
a value, [latex]r[/latex], between [latex]–1[/latex] and [latex]1[/latex] that indicates the degree of linear correlation of variables or how closely a regression line fits a data set.
cross-multiplication
A method for solving rational equations set up as a proportion by multiplying terms across the equal sign.
decreasing function
a function is decreasing in some open interval if [latex]f\left(b\right)a[/latex]
decreasing linear function
a function with a negative slope: If [latex]m<0, \text{then }f\left(x\right)=mx+b[/latex] is decreasing.
degree
the highest power of the variable that occurs in a polynomial
dependent system
A system of linear equations in which the two equations represent the same line; there are an infinite number of solutions to a dependent system
dependent variable
an output variable
Descartes’ Rule of Signs
a rule that determines the maximum possible numbers of positive and negative real zeros based on the number of sign changes of [latex]f\left(x\right)[/latex] and [latex]f\left(-x\right)[/latex]
difference of squares
the binomial that results when a binomial is multiplied by a binomial with the same terms, but the opposite sign
direct variation
the relationship between two variables that are a constant multiple of each other; as one quantity increases, so does the other
directrix
a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant
discriminant
the value under the radical in the quadratic formula, [latex]b^2-4ac[/latex], which tells whether the quadratic has real or complex roots
distance formula
a formula that can be used to find the length of a line segment if the endpoints are known
distributive property
the product of a factor times a sum is the sum of the factor times each term in the sum; in symbols, [latex]a\cdot \left(b+c\right)=a\cdot b+a\cdot c[/latex]
Division Algorithm
given a polynomial dividend [latex]f\left(x\right)[/latex] and a non-zero polynomial divisor [latex]d\left(x\right)[/latex] where the degree of [latex]d\left(x\right)[/latex] is less than or equal to the degree of [latex]f\left(x\right)[/latex], there exist unique polynomials [latex]q\left(x\right)[/latex] and [latex]r\left(x\right)[/latex] such that [latex]f\left(x\right)=d\left(x\right)q\left(x\right)+r\left(x\right)[/latex] where [latex]q\left(x\right)[/latex] is the quotient and [latex]r\left(x\right)[/latex] is the remainder. The remainder is either equal to zero or has degree strictly less than [latex]d\left(x\right)[/latex].
diverge
a series is said to diverge if the sum is not a real number
domain
the set of all possible input values for a relation
doubling time
the time it takes for a quantity to double
ellipse
The set of all points [latex]\left(x,y\right)[/latex] in a plane such that the sum of their distances from two fixed points is a constant
end behavior
the behavior of the graph of a function as the input decreases without bound and increases without bound
entry
an element, coefficient, or constant in a matrix
equation
a mathematical statement indicating that two expressions are equal
equation in two variables
a mathematical statement, typically written in x and y, in which two expressions are equal
even function
a function whose graph is unchanged by horizontal reflection, [latex]f\left(x\right)=f\left(-x\right)[/latex], and is symmetric about the [latex]y\text{-}[/latex] axis
event
any subset of a sample space
excluded values
Values that make the denominator in a rational expression equal to zero, which must be excluded from the solution set.
experiment
an activity with an observable result
explicit formula
a formula that defines each term of a sequence in terms of its position in the sequence
exponent
in exponential notation, the raised number or variable that indicates how many times the base is being multiplied
exponential growth
a model that grows by a rate proportional to the amount present
exponential notation
a shorthand method of writing products of the same factor
extraneous solution
a solution introduced while solving an equation that does not satisfy the conditions of the original equation
extrapolation
predicting a value outside the domain and range of the data
factor by grouping
a method for factoring a trinomial of the form [latex]a{x}^{2}+bx+c[/latex] by dividing the x term into the sum of two terms, factoring each portion of the expression separately, and then factoring out the GCF of the entire expression
Factor Theorem
k is a zero of polynomial function [latex]f\left(x\right)[/latex] if and only if [latex]\left(x-k\right)[/latex] is a factor of [latex]f\left(x\right)[/latex]
finite sequence
a function whose domain consists of a finite subset of the positive integers [latex]\left\{1,2,\dots n\right\}[/latex] for some positive integer [latex]n[/latex]
focal diameter (latus rectum)
the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola
foci
Plural of focus
focus (of an ellipse)
One of the two fixed points on the major axis of an ellipse such that the sum of the distances from these points to any point [latex]\left(x,y\right)[/latex] on the ellipse is a constant
focus (of a parabola)
a fixed point in the interior of a parabola that lies on the axis of symmetry
formula
an equation expressing a relationship between constant and variable quantities
function
a relation in which each input value yields a unique output value
fundamental counting principle
if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Multiplication Principle
Fundamental Theorem of Algebra
a polynomial function with degree greater than 0 has at least one complex zero
Gaussian elimination
using elementary row operations to obtain a matrix in row-echelon form
general form of a quadratic function
the function that describes a parabola, written in the form [latex]f\left(x\right)=a{x}^{2}+bx+c[/latex], where [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are real numbers and [latex]a\ne 0[/latex].
geometric sequence
a sequence in which the ratio of a term to a previous term is a constant
geometric series
the sum of the terms in a geometric sequence
global maximum
highest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\ge f\left(x\right)[/latex] for all [latex]x[/latex].
global minimum
lowest turning point on a graph; [latex]f\left(a\right)[/latex] where [latex]f\left(a\right)\le f\left(x\right)[/latex] for all [latex]x[/latex].
graph in two variables
the graph of an equation in two variables, which is always shown in two variables in the two-dimensional plane
greatest common factor
the largest polynomial that divides evenly into each polynomial
half-life
the length of time it takes for a substance to exponentially decay to half of its original quantity
horizontal asymptote
a horizontal line [latex]y=b[/latex] where the graph approaches the line as the inputs increase or decrease without bound.
horizontal compression
a transformation that compresses a function’s graph horizontally, by multiplying the input by a constant [latex]b>1[/latex]
horizontal line
a line defined by [latex]f\left(x\right)=b[/latex] where [latex]b[/latex] is a real number. The slope of a horizontal line is [latex]0[/latex].
horizontal line test
a method of testing whether a function is one-to-one by determining whether any horizontal line intersects the graph more than once
horizontal reflection
a transformation that reflects a function’s graph across the y-axis by multiplying the input by [latex]-1[/latex]
horizontal shift
a transformation that shifts a function’s graph left or right by adding a positive or negative constant to the input
horizontal stretch
a transformation that stretches a function’s graph horizontally by multiplying the input by a constant [latex]0
hyperbola
The set of all points [latex]\left(x,y\right)[/latex] in a plane such that the difference of the distances between [latex]\left(x,y\right)[/latex] and the foci is a positive constant
identity matrix
a square matrix containing ones down the main diagonal and zeros everywhere else; it acts as a 1 in matrix algebra
identity property of addition
there is a unique number, called the additive identity, 0, which, when added to a number, results in the original number; in symbols, [latex]a+0=a[/latex]
identity property of multiplication
there is a unique number, called the multiplicative identity, [latex]1[/latex], which, when multiplied by a number, results in the original number; in symbols, [latex]a\cdot 1=a[/latex]
imaginary number
a number in the form [latex]bi[/latex] where [latex]i=\sqrt{-1}[/latex]
inconsistent system
A system of linear equations with no common solution because they represent parallel lines, which have no point or line in common
increasing function
a function is increasing in some open interval if [latex]f\left(b\right)>f\left(a\right)[/latex] for any two input values [latex]a[/latex] and [latex]b[/latex] in the given interval where [latex]b>a[/latex]
increasing linear function
a function with a positive slope: If [latex]m>0, \text{then }f\left(x\right)=mx+b[/latex] is increasing.
independent system
A system of linear equations with exactly one solution pair [latex]\left(x,y\right)[/latex]
independent variable
an input variable
index
the number above the radical sign indicating the nth root
index of summation
in summation notation, the variable used in the explicit formula for the terms of a series and written below the sigma with the lower limit of summation
infinite sequence
a function whose domain is the set of positive integers
infinite series
the sum of the terms in an infinite sequence
input
each object or value in a domain that relates to another object or value by a relationship known as a function
integers
the set consisting of the natural numbers, their opposites, and [latex]0[/latex]: [latex]\{\dots ,-3,-2,-1,0,1,2,3,\dots \}[/latex]
intercepts
the points at which the graph of an equation crosses the x-axis and the y-axis
Intermediate Value Theorem
for two numbers [latex]a[/latex] and [latex]b[/latex] in the domain of [latex]f[/latex], if [latex]a
interpolation
predicting a value inside the domain and range of the data
interval
an interval describes a set of numbers where a solution falls
interval notation
a mathematical statement that describes a solution set and uses parentheses or brackets to indicate where an interval begins and ends
inverse function
for any one-to-one function [latex]f\left(x\right)[/latex], the inverse is a function [latex]{f}^{-1}\left(x\right)[/latex] such that [latex]{f}^{-1}\left(f\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]f[/latex]; this also implies that [latex]f\left({f}^{-1}\left(x\right)\right)=x[/latex] for all [latex]x[/latex] in the domain of [latex]{f}^{-1}[/latex]
inverse property of addition
for every real number [latex]a[/latex], there is a unique number, called the additive inverse (or opposite), denoted [latex]-a[/latex], which, when added to the original number, results in the additive identity, 0; in symbols, [latex]a+\left(-a\right)=0[/latex]
inverse property of multiplication
for every non-zero real number [latex]a[/latex], there is a unique number, called the multiplicative inverse (or reciprocal), denoted [latex]\dfrac{1}{a}[/latex], which, when multiplied by the original number, results in the multiplicative identity, 1; in symbols, [latex]a\cdot \dfrac{1}{a}=1[/latex]
inverse variation
the relationship between two variables in which the product of the variables is a constant
inversely proportional
a relationship where one quantity is a constant divided by the other quantity; as one quantity increases, the other decreases
invertible function
any function that has an inverse function
irrational numbers
the set of all numbers that are not rational; they cannot be written as either a terminating or repeating decimal; they cannot be expressed as a fraction of two integers
joint variation
a relationship where a variable varies directly or inversely with multiple variables
leading coefficient
the coefficient of the leading term
leading term
the term containing the highest degree
least common denominator
the smallest multiple that two denominators have in common
least squares regression
a statistical technique for fitting a line to data in a way that minimizes the differences between the line and data values
Linear Factorization Theorem
allowing for multiplicities, a polynomial function will have the same number of factors as its degree, and each factor will be in the form [latex]\left(x-c\right)[/latex] where c is a complex number
linear function
a function with a constant rate of change that is a polynomial of degree [latex]1[/latex] whose graph is a straight line
linear inequality
similar to a linear equation except that the solutions will include an interval of numbers
local extrema
collectively, all of a function’s local maxima and minima
local maximum
a value of the input where a function changes from increasing to decreasing as the input value increases.
local minimum
a value of the input where a function changes from decreasing to increasing as the input value increases.
logarithm
the exponent to which [latex]b[/latex] must be raised to get [latex]x[/latex]; written [latex]y={\mathrm{log}}_{b}\left(x\right)[/latex]
logistic growth model
a function of the form [latex]f\left(x\right)=\frac{c}{1+a{e}^{-bx}}[/latex] where [latex]\frac{c}{1+a}[/latex] is the initial value, c is the carrying capacity, or limiting value, and b is a constant determined by the rate of growth
lower limit of summation
the number used in the explicit formula to find the first term in a series
main diagonal
entries from the upper left corner diagonally to the lower right corner of a square matrix
major Axis
The longer of the two axes of an ellipse
matrix
a rectangular array of numbers
midpoint formula
a formula to find the point that divides a line segment into two parts of equal length
minor Axis
The shorter of the two axes of an ellipse
model breakdown
when a model no longer applies after a certain point
monomial
a polynomial containing one term
multiplication principle
if one event can occur in [latex]m[/latex] ways and a second event can occur in [latex]n[/latex] ways after the first event has occurred, then the two events can occur in [latex]m\times n[/latex] ways; also known as the Fundamental Counting Principle
multiplicative inverse of a matrix
a matrix that, when multiplied by the original, equals the identity matrix
multiplicity
the number of times a given factor appears in the factored form of the equation of a polynomial; if a polynomial contains a factor of the form [latex]{\left(x-h\right)}^{p}[/latex], [latex]x=h[/latex] is a zero of multiplicity [latex]p[/latex].
mutually exclusive events
events that have no outcomes in common
natural logarithm
the exponent to which the number [latex]e[/latex] must be raised to get [latex]x[/latex]; [latex]{\mathrm{log}}_{e}\left(x\right)[/latex] is written as [latex]\mathrm{ln}\left(x\right)[/latex]
natural numbers
the set of counting numbers: [latex]\{1,2,3,\dots \}[/latex]
Newton’s Law of Cooling
the scientific formula for temperature as a function of time as an object’s temperature is equalized with the ambient temperature
[latex]n[/latex] factorial
the product of all the positive integers from 1 to [latex]n[/latex]
nominal rate
the yearly interest rate earned by an investment account, also called annual percentage rate
[latex]n[/latex]th partial sum
the sum of the first [latex]n[/latex] terms of a sequence
[latex]n[/latex]th term of a sequence
a formula for the general term of a sequence
odd function
a function whose graph is unchanged by combined horizontal and vertical reflection, [latex]f\left(x\right)=-f\left(-x\right)[/latex], and is symmetric about the origin
one-to-one function
a function for which each value of the output is associated with a unique input value
order of magnitude
the power of ten when a number is expressed in scientific notation with one non-zero digit to the left of the decimal
order of operations
a set of rules governing how mathematical expressions are to be evaluated, assigning priorities to operations
ordered pair
a pair of numbers indicating horizontal displacement and vertical displacement from the origin; also known as a coordinate pair, [latex]\left(x,y\right)[/latex]
origin
the point where the two axes cross in the center of the plane, described by the ordered pair [latex]\left(0,0\right)[/latex]
outcomes
the possible results of an experiment
output
each object or value in the range that is produced when an input value is entered into a function
parabola
the set of all points [latex]\left(x,y\right)[/latex] in a plane that are the same distance from a fixed line, called the directrix, and a fixed point (the focus) not on the directrix
parallel lines
two or more lines with the same slope
partial fractions
the individual fractions that make up the sum or difference of a rational expression before combining them into a simplified rational expression
partial fraction decomposition
the process of returning a simplified rational expression to its original form, a sum or difference of simpler rational expressions
perfect square trinomial
the trinomial that results when a binomial is squared
perimeter
in linear units, the perimeter formula is used to find the linear measurement, or outside length and width, around a two-dimensional regular object; for a rectangle: [latex]P=2L+2W[/latex]
permutation
a selection of objects in which order matters
perpendicular lines
two lines that intersect at right angles and have slopes that are negative reciprocals of each other
piecewise function
a function in which more than one formula is used to define the output
point-slope form
the equation of a linear function of the form [latex]y-{y}_{1}=m\left(x-{x}_{1}\right)[/latex]
polynomial
a sum of terms each consisting of a variable raised to a nonnegative integer power
polynomial function
a function that consists of either zero or the sum of a finite number of non-zero terms, each of which is a product of a number, called the coefficient of the term, and a variable raised to a non-negative integer power.
polynomial equation
an equation containing a string of terms including numerical coefficients and variables raised to whole-number exponents
power function
a function that can be represented in the form [latex]f\left(x\right)=a{x}^{n}[/latex] where a is a constant, the base is a variable, and the exponent is n, is a smooth curve represented by a graph with no sharp corners
power rule for logarithms
a rule of logarithms that states that the log of a power is equal to the product of the exponent and the log of its base
principal nth root
the number with the same sign as [latex]a[/latex] that when raised to the nth power equals [latex]a[/latex]
principal square root
the nonnegative square root of a number [latex]a[/latex] that, when multiplied by itself, equals [latex]a[/latex]
probability
a number from [latex]0[/latex] to [latex]1[/latex] indicating the likelihood of an event
probability model
a mathematical description of an experiment listing all possible outcomes and their associated probabilities
product rule for logarithms
a rule of logarithms that states that the log of a product is equal to a sum of logarithms
profit function
The profit function is written as [latex]P\left(x\right)=R\left(x\right)-C\left(x\right)[/latex], revenue minus cost
projectile motion
motion of an object thrown or launched near Earth’s surface
Pythagorean Theorem
a theorem that states the relationship among the lengths of the sides of a right triangle, used to solve right triangle problems
quadrant
one quarter of the coordinate plane, created when the axes divide the plane into four sections
quadratic equation
an equation containing a second-degree polynomial; can be solved using multiple methods
quadratic formula
a formula that will solve all quadratic equations
quadratic function
A polynomial function of degree 2
quotient rule for logarithms
a rule of logarithms that states that the log of a quotient is equal to a difference of logarithms
radical
the symbol used to indicate a root
radical equation
an equation containing at least one radical term where the variable is part of the radicand
radical expression
an expression containing a radical symbol
radicand
the number under the radical symbol
range
the set of output values that result from the input values in a relation
rate
speed or frequency at which something occurs
rate of change
the change of an output quantity relative to the change of the input quantity
rational equation
An equation that contains at least one rational expression, where the variable appears in at least one of the denominators.
rational expression
The ratio or quotient of two polynomials, e.g., \( \frac{x+1}{x^2-4} \), \( \frac{1}{x-3} \), or \( \frac{4}{x^2+x-2} \).
rational function
a function that can be written as the ratio of two polynomials
rational numbers
the set of all numbers of the form [latex]\dfrac{m}{n}[/latex], where [latex]m[/latex] and [latex]n[/latex] are integers and [latex]n\ne 0[/latex]. Any rational number may be written as a fraction or a terminating or repeating decimal.
Rational Zero Theorem
the possible rational zeros of a polynomial function have the form [latex]\frac{p}{q}[/latex] where p is a factor of the constant term and q is a factor of the leading coefficient
real number line
a horizontal line used to represent the real numbers. An arbitrary fixed point is chosen to represent 0; positive numbers lie to the right of 0 and negative numbers to the left.
real numbers
the sets of rational numbers and irrational numbers taken together
recursive formula
a formula that defines each term of a sequence using previous term(s)
relation
a set of ordered pairs
Remainder Theorem
if a polynomial [latex]f\left(x\right)[/latex] is divided by [latex]x-k[/latex] , then the remainder is equal to the value [latex]f\left(k\right)[/latex]
removable discontinuity
a single point at which a function is undefined that, if filled in, would make the function continuous; it appears as a hole on the graph of a function
revenue function
The function that is used to calculate revenue, simply written as [latex]R=xp[/latex], where [latex]x=[/latex] quantity and [latex]p=[/latex] price
row
a set of numbers aligned horizontally in a matrix
row-echelon form
after performing row operations, the matrix form that contains ones down the main diagonal and zeros at every space below the diagonal
row-equivalent
two matrices [latex]A[/latex] and [latex]B[/latex] are row-equivalent if one can be obtained from the other by performing basic row operations
row operations
adding one row to another row, multiplying a row by a constant, interchanging rows, and so on, with the goal of achieving row-echelon form
sample space
the set of all possible outcomes of an experiment
scalar multiple
an entry of a matrix that has been multiplied by a scalar
scientific notation
a shorthand notation for writing very large or very small numbers in the form [latex]a\times {10}^{n}[/latex] where [latex]1\le |a|<10[/latex] and [latex]n[/latex] is an integer
sequence
a function whose domain is a subset of the positive integers
series
the sum of the terms in a sequence
set-builder notation
a method of describing a set by a rule that all of its members obey; it takes the form [latex]\left\{x|\text{statement about }x\right\}[/latex]
slope
the change in [latex]y[/latex]–values over the change in [latex]x[/latex]–values
slope-intercept form
the equation of a linear function of the form [latex]f\left(x\right)=mx+b[/latex]
solution set
the set of all ordered pairs or triples that satisfy all equations in a system of equations
square root property
one of the methods used to solve a quadratic equation in which the [latex]{x}^{2}[/latex] term is isolated so that the square root of both sides of the equation can be taken to solve for x
standard form of a quadratic function
the function that describes a parabola, written in the form [latex]f\left(x\right)=a{\left(x-h\right)}^{2}+k[/latex], where [latex]\left(h,\text{ }k\right)[/latex] is the vertex.
substitution method
An algebraic technique used to solve systems of linear equations in which one of the two equations is solved for one variable and then substituted into the second equation to solve for the second variable
summation notation
a notation for series using the Greek letter sigma; it includes an explicit formula and specifies the first and last terms in the series
synthetic division
a shortcut method that can be used to divide a polynomial by a binomial of the form [latex]x – k[/latex]
system of linear equations
A set of two or more equations in two or more variables that must be considered simultaneously
term
a number in a sequence
term of a polynomial
any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial of the form [latex]{a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
term of a polynomial function
any [latex]{a}_{i}{x}^{i}[/latex] of a polynomial function in the form [latex]f\left(x\right)={a}_{n}{x}^{n}+\dots+{a}_{2}{x}^{2}+{a}_{1}x+{a}_{0}[/latex]
transverse Axis
The axis of a hyperbola that includes the foci and has the vertices as its endpoints
trinomial
a polynomial containing three terms
turning point
the location where the graph of a function changes direction
union of two events
the event that occurs if either or both events occur
upper limit of summation
the number used in the explicit formula to find the last term in a series
variable
a quantity that may change value
varies directly
a relationship where one quantity is a constant multiplied by the other quantity
varies inversely
a relationship where one quantity is a constant divided by the other quantity
vertex
The highest or lowest point of a parabola [latex](h,k)[/latex]
vertical asymptote
a vertical line [latex]x=a[/latex] where the graph tends toward positive or negative infinity as the inputs approach [latex]a[/latex]
vertical compression
a function transformation that compresses the function’s graph vertically by multiplying the output by a constant [latex]0
vertical line
a line defined by [latex]x=a[/latex] where a is a real number. The slope of a vertical line is undefined.
vertical line test
a method of testing whether a graph represents a function by determining whether a vertical line intersects the graph no more than once
vertical reflection
a transformation that reflects a function’s graph across the x-axis by multiplying the output by [latex]-1[/latex]
vertical shift
a transformation that shifts a function’s graph up or down by adding a positive or negative constant to the output
vertical stretch
a transformation that stretches a function’s graph vertically by multiplying the output by a constant [latex]a>1[/latex]
volume
in cubic units, the volume measurement includes length, width, and depth: [latex]V=LWH[/latex]
whole numbers
the set consisting of [latex]0[/latex] plus the natural numbers: [latex]\{0,1,2,3,\dots \}[/latex]
x-axis
the common name of the horizontal axis on a coordinate plane; a number line increasing from left to right
x-coordinate
the first coordinate of an ordered pair, representing the horizontal displacement and direction from the origin
x-intercept
the point where a graph intersects the x-axis; an ordered pair with a y-coordinate of zero
y-axis
the common name of the vertical axis on a coordinate plane; a number line increasing from bottom to top
y-coordinate
the second coordinate of an ordered pair, representing the vertical displacement and direction from the origin
y-intercept
a point where a graph intercepts the y-axis; an ordered pair with an x-coordinate of zero
zero-product property
the property that formally states that multiplication by zero is zero so that each factor of a quadratic equation can be set equal to zero to solve equations