Unit 1 Foundations of Algebra
absolute value
the distance from zero on the number line
additive identity
the number 0 (for the real numbers)
additive inverse
the opposite or negative of a number; the sum of a number and its additive inverse is zero
algebraic expression
a mathematical expression containing one or more variables
associative property
a property of the real numbers which states that how numbers are grouped in a sum or product does not change the value
base
the number which is repeated as a factor in an exponential expression
coefficient
the constant preceding the variables in a product
commutative property
a property of the real numbers which states that the order in which numbers are added or multiplied does not change the value
constant
a numerical value; a term containing no variables
distributive property
a(b + c) = ab + ac, or a(b - c) = ab - ac
evaluate
to find the numerical value of
exponent
the (superscript) number in an exponential expression; it tells how many times a factor is repeated
expression
a mathematical phrase that cannot be determined true or false
integer
a number belonging to the set made up of the whole numbers and their opposites
irrational number
a number with an infinite decimal representation that does not repeat
like terms
terms that have the same variable(s), with each variable raised to the same exponent
multiplicative inverse
the reciprocal of a number; if x is a number other than zero, then 1/x is the multiplicative inverse; the product of a number and its multiplicative inverse is 1
natural number
a number belonging to the set made up of the numbers that are used to count: 1, 2, 3, and so on
rational number
a number that can be written as a ratio of two integers in the form A/B with B ≠0
substitution
replacing a quantity with its equal
term
a number or a variable, or an indicated product of a number and variable(s)
variable
a letter or symbol used to represent an unknown
whole number
a number belonging to the set made up of zero and the natural numbers
Unit 2 Linear Equations
additive identity
the number 0 (for the real numbers)
distributive property
a(b + c) = ab + ac or a(b - c) = ab - ac
equation
a mathematical sentence whose verb is 'equal' (=)
identity
an equation that is true for all values of the variable
interest
a payment for the use of money borrowed or invested
like terms
terms that have the same variable(s), with each variable raised to the same exponent
literal equation
an equation containing more than one variable
multiplicative identity
the number 1 (for the real numbers)
multiplicative inverse
the reciprocal of a number; if x is a number other than zero, then 1/x is the multiplicative inverse; the product of a number and its multiplicative inverse is 1
open sentence
an equation that contains a variable or variables
principal
the amount of money invested or borrowed at a certain rate of interest
solution
a value or values of the variable that make an algebraic sentence true
symmetric property
when the sides of an equation are interchanged, an equivalent equation is formed; if a = b, then b = a
variable
a letter or symbol used to represent an unknown
Unit 3 Functions
abscissa
the first number (or x value) of an ordered pair
absolute value
the distance that a number is from zero on the number line; always positive
arithmetic sequence
a sequence in which a fixed amount is added on to get the next term
common difference
the fixed amount added on to get to the next term in an arithmetic sequence
constant of variation
the value of k in a direct variation y = kx
coordinate plane
a way of locating points in a plane that consists of a horizontal and a vertical number line intersecting at the zeros
coordinates
an ordered pair, (x, y), that describes the location of a point in the coordinate plane
dependent variable
the variable representing the second elements of the ordered pairs in a function; the outputs
direct variation
a function of the form y = kx where k is not zero
domain
the set of first coordinates in a relation
f-inverse
The inverse function of f is written f -1 and pronounced "f-inverse."
function
a relation in which each first coordinate is paired with exactly one second coordinate
independent variable
the variable representing the first elements of the ordered pairs in a function; the inputs
inverse function
a function f -1 that switches the input and output of the original function f. So, if f( x) = y then f -1( y) = x.
linear equation
an open sentence whose graph is a line; the general form for such an equation is Ax + By + C = 0
ordinate
the second number (or y value) of an ordered pair
origin
the point of intersection, (0, 0), of the axes in the coordinate plane
parallel lines
lines that never intersect
piecewise-defined function
a function that has different rules for different parts of its domain
perpendicular lines
lines that intersect to form right (90 degree) angles
quadrant
region in the coordinate plane
range
the set of second coordinates in a relation
rate of change
how the dependent variable changes with respect to the independent variable
relation
a set of ordered pairs
sequence
a set of numbers that follow a pattern, with a specific first number
set
a collection or group of objects indicated by braces, { }
slope
the rate of change of a line; change in y over change in x; rise over run
slope-intercept form
y = mx + b form of a linear equation
solution
a value or values of the variable that make an algebraic sentence true
stepwise function
a special type of a piecewise-defined function in which each piece is a linear function
term
an individual quantity or number in a sequence
translation
a shift or slide of a graph in the coordinate plane
x-axis
the horizontal axis in the coordinate plane
x-intercept
the point where the line crosses the x-axis
y-axis
the vertical axis in the coordinate plane
y-intercept
the point where the line crosses the y-axis
Unit 4 Inequalities
absolute value
the distance from zero on the number line
compound inequality
a statement formed by two or more inequalities
element
a member of a set
intersection
the intersection of sets A and B is defined as any elements that are in both set A and set B
linear inequality
an open sentence of the form Ax + By + C < 0 or Ax + By + C > 0
maximum
the largest value
minimum
the smallest value
null set
a set containing no elements; also called the empty set
set
a collection or group of objects indicated by braces, { }
solution
a value or values of the variable that make an algebraic sentence true
subset
set A is a subset of set B if all of the elements of set A are contained in set B or it is the empty set
union
the union of sets A and B is defined as any elements that are in either set A or set B
Unit 5 Linear Terms
coefficient
the constant preceding the variables in a product
consistent equations
A system of linear equations that contain at least one common point.
dependent equations
A system of linear equations that rely on each other for the algebraic or graphic form of the equation.
determinant
the value of: (row 1, column 1)(row 2 column 2) - (row 1, column 2)(row 2, column 1) in a 2 by 2 matrix
equivalent equations
equations having all common solutions
inconsistent equations
A system of linear equations that do not contain any common points.
independent equations
A system of linear equations that do not rely on each other for the algebraic or graphic form of the equation.
infinitely many solutions
A set of linear equations that coincide and share every point as a point of intersection. Also known as a dependent and consistent solution.
linear inequality
an open sentence of the form Ax + By + C < 0 or Ax + By + C > 0
matrix
a rectangular array made up of rows and columns
no solution
A set of parallel lines that will never share a point of intersection. Considered to be an inconsistent solution.
one solution
A set of linear equations that share a common point known as the point of intersection (x,y). The solution, (x,y) is an independent and consistent solution.
standard form
the form Ax + By = C of a linear equation
substitute
replace a quantity with its equal
system determinant
the determinant found when column 1 consists of the x-coefficients and column 2 consists of the y-coefficients of a linear system
x-determinant
the determinant found when column 1 consists of the constants and column 2 consists of the y-coefficients of a linear system
y-determinant
the determinant found when column 1 consists of the x-coefficients and column 2 consists of the constants of a linear system