the distance from the number to zero on a number line
Example: |3| = 3 or |-3| = 3
absolute value function
a function written in the vertex form f(x) = a |x – h| + k, where a ≠ 0; the graph of an absolute value function is a “V” shape
asymptote
a line that a curved graph gets closer to without every touching
at least
must be more than
axis of symmetry
for a quadratic function, a vertical line that includes the vertex of a quadratic function resulting in the two sides of the graph look like mirror images of each other
base
in a logarithm, the base is the number that is raised to a power
If nx = a, the logarithm of a, n is the base.
binomial
a polynomial with two terms
Examples: (x - 2) or (3x + 5)
change of base formula
To change a logarithmic expression with base b to a common log (base 10), use the formula:
log_{b} (ARGUMENT) =
log (ARGUMENT)log b
circle
the locus of points that are a fixed distance from a given point
coefficient
the number multiplied times a product of variables or powers of variables in a term
compensate
the act of making equal
completing the square
a method of symbolic manipulation in which a polynomial can be rewritten to include a binomial that is squared
complex number
a number of the form a + bi where a and b are real numbers, and b ≠ 0
compounded monthly
when interest is earned not only on the original principal but also on the accumulated interest of prior months
compression
a transformation that pushes the points of a graph vertically toward the x-axis
congruent
having the same size and same shape
conjugate axis
the line through the center of the hyperbola that is perpendicular to the line through the foci
constant
a term or expression with no variables; a value that does not change
continuous
a set of points without breaks
critical attribute
characteristics of a quadratic function that set it apart from others, including: x-intercepts, y-intercepts, vertex, and the axis of symmetry
decreasing
denotes the direction in a parabola, the y-coordinates are decreasing while the x-coordinates are increasing
degree
the highest exponent in an equation
dependent variable
a variable (usually “y”) whose value is found by using the value of the independent variable (usually “x”)
depreciates
losing value over time
dimensions of a matrix
When listing the dimensions of a matrix, give the number of rows followed by the number of columns.
direct variation
a relationship in which the ratio between two variables is constant
If k = yx or y = kx, where k is a non-zero constant, then y varies directly with x.
discontinuities
places in the graph of a function where the graph is not continuous; these can be asymptotes or holes
discrete
a set of individual points
distributive property
a property of real numbers that states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products
x(a + b) = ax + bx
domain
the set of input values of a function or relation
elimination
the act of adding two equations of a system together in such a way that one of the variables is removed or eliminated from the resulting equation
ellipse
a closed, symmetric curve shaped like an oval, which can be formed by intersecting a cone with a plane that is not parallel or perpendicular to the cone's base
The set of points for which the sum of the distances to two foci within the curve are congruent.
equal
the same value
equation
a mathematical sentence that equates two expressions; an equation contains an equal sign, =
exponent
says how many times to use the number in a multiplication
In this example: 8^{2} = 8 × 8 = 64 orx^{2} = x × x.
exponential
an expression written using exponents [e.g., y = 2^{x}]
exponential decay
for f(x) = a * B^{x+c} + d, if a > 0 and 0 < B < 1, then the y-values decrease (or decay) as the x-values increase
The rate of decay is proportional to the y-value, so as the y-values get smaller, the graph flattens out (or approaches a horizontal asymptote).
exponential growth
for f(x) = a * B^{x+c} + d, if a > 0 and B > 1, then the y-values increase (or grow) as the x-values increase
The rate of decay is proportional to the y-value, so as the y-values get larger, the graph grows without bound (or exponentially).
expression
a mathematical sentence that shows a relationship among real numbers and variables; an expression does not contain a symbol implying equality or inequality
extraneous roots
a solution of a modified equation that is NOT a solution to the original equation
dxtraneous solution
answer that must be discarded because of a domain restriction or similar problem
focal radius
distance from the center of a hyperbola to each focal point along the major axis
focus
a fixed point on the interior of a parabola
The distance from the focus to the parabola is equal to the distance from the directrix to the parabola.
formulate
to develop or create a formula or equation
function
a relation that matches each element in the domain with exactly one element of the range
holes
points where the graph of a function does not exist
horizontal asymptote
aline that a graph approaches more and more closely
horizontal shift
movement of a graph to the right or left
A function f(x) can be moved “h” units right/left by performing the operation f(x ± h).
hyperbola
a plane curve having two branches, formed by the intersection of a plane with both halves of a right circular cone at an angle parallel to the axis of the cone
It is the locus of points for which the difference of the distances from two given points is a constant.
identity matrix
a square matrix with a diagonal of 1's and the remaining elements are 0's
imaginary number
a value that includes the square root of a negative number
imaginary root
A quadratic equation is said to have imaginary roots when the quadratic formula yields imaginary or non-real answers. Also, the graph of a function with imaginary roots will not touch the x-axis.
increasing
denotes the direction in a parabola, the y-coordinates are increasing while the x-coordinates are increasing
independent variable
the input of a function (usually “x”) whose value is used to find the value of the dependent variable (usually “y”), values can be freely chosen
index of a radical
the nth root of x where n is the index is represented by the radical; radical √x has an index of 2 and is often called the square root of x
inequality
a number sentence where one quantity can be greater or less than another quantity, a number sentences with inequality symbols
infinitely many solutions
a system in which there are an infinite number of ordered pairs that satisfy all of the equations; also referred to as a dependent system
infinite
an unlimited number
infinity
going on forever, of not having an end, of having no boundary
integer
the set of positive and negative whole numbers {…-2, -1, 0, 1, 2 …}
integral
in the form of an integer (positive and negative whole numbers)
inverse
a function that undoes what the original function did
inverse of a function
a function obtained by exchanging the input and output values of a one-to-one function
inverse of a matrix
a second matrix that, when multiplied by the first matrix, will always result in the identity matrix
inverse variation
a relationship in which the product of two variables is constant
If xy = k or . y = kxwhere k is a non-zero constant, then y varies inversely with x.
irrational number
a number that cannot be written in fractional form, a non-terminating (does not end) and non-repeating decimal value, examples include all non-perfect square roots (√2, √3, etc. and π)
isolate
get something by itself
line of best fit (or "trend" line)
a straight line that best represents the data on a scatter plot; may pass through some of the points, none of the points, or all of the points
logarithm
a power to which a base, [actually 10] must be raised to produce a given number
If n^{x} = a, the logarithm of a, with n as the base, is x.
logarithmic function
function written in the simple base form y = log_{B}(x)
These functions can be rewritten without logarithms in the equivalent statement: x = B^{y} Only positive real numbers have real logarithms. You cannot take the logarithm of a negative number. For instance, y = log_{2}(-4) is undefined since the equivalent statement: -4 = 2^{y} has no solution. You cannot raise 2 to any power (y) to generate -4 as the answer.
major axis
the line that runs through the foci, center, and vertices of a hyperbola
matrix
a rectangular arrangement of numbers
"Matrices" is the plural form of "matrix".
maximum value
the highest point on the graph of the quadratic function
minimum value
the lowest point on the graph of the quadratic function
minor axis
the line through the center of a hyperbola that is perpendicular to the major axis
multiplying binomials
Use the FOIL method to multiply binomials.
no solution
when there isn’t any value that will make the equation true
one-to-one function
a function where every element of the range corresponds to exactly one element of the domain and passes both the vertical line test and the horizontal line test
parabola
a U-shaped curve that is the graph of a quadratic function
parameter
a constant in a function or equation that may be changed
parent function
the simplest function for a family of functions
perfect square trinomial
a trinomial that is the square of a binomial
Perfect squares factor into either the square of a sum such as (x + 3)^{2,} or the square of a difference such as (x - 9)^{2}.
piecewise
a function that is a combination of pieces of two or more other functions
polynomial
an expression with one or more terms containing variables, real numbers, or products of one or more variables and a real number with whole-number exponents
polynomial form
an equation that is written as a polynomial; y = ax^{2} + bx + c is the polynomial form of a quadratic equation
principal square root
the unique nonnegative square root of a nonnegative real number
quadratic
a second degree equation, which can be written in general form y = Ax^{2} + Bx + C
quadratic expression/quadratic function
a polynomial function whose highest exponent is 2; the graph is a parabola
The general form is: f(x) = ax^{2} + bx + c.
quadratic formula
the formula which gives solutions or roots for equations of the
form ax^{2} + bx + c (a ≠ 0)
radical
a root of a quantity, e
Example: √x
radicand
the number or expression under a radical sign
radius
a line segment between the center and a point on the circle
range
the set of values of the dependent variable for which a function is defined
rational function
a function that can be written as a quotient of two polynomial expressions where the denominator has degree 1 or higher
There needs to be a variable in the denominator to be a rational function.
rational number
any number that can be expressed as a ratio (fraction)
real number
every point on the number line; can be either a rational or irrational number
Reduced Row Echelon Form (RREF)
a matrix form in which each row is reduced to a 1 along the diagonal
All entries above and below the diagonal are 0’s.
reflection
a transformation "flips" across a line (called the line of reflection)
Only the location of the graph changes, the shape stays the same.
relation
a set of ordered pairs
restricted domain
the domain of a function is restricted when there are values of x that cannot be used, when you substitute in an x - value the y-value is undefined or imaginary
roots
the solution(s) of a quadratic equation
square
to raise a number or expression to the 2nd power
square root function
a function whose rule contains a variable beneath a square-root sign
simulate
an activity that represents a real-world situation without the need to actually perform the real-world situation
solution
roots, zeros, x-intercepts, or values where the function equals zero
solution to a system of equations
the point or points of intersection for all equations in the system
standard form
a quadratic equation written in the form: y = ax^{2} + bx + c, where a ≠ 0
stretch
a transformation that pulls the points of a graph vertically away from the x-axis
substitution
the act of replacing a variable with a number or expression equal in value
symmetric
if a graph is symmetric, two parts of the graph are congruent to each other
system of inequalities
two or more inequalities containing common variables
system of linear equations
two or more equations involving the same variables and worked upon together
transformation
a change in the shape or location of a graph
translation
a change in only the location of a graph; all of the points in the graph move the same distance in the same direction and the shape of the graph stays the same
transverse axis
the line through the center of the hyperbola that includes the foci
trinomial
a polynomial with three terms
variable
(usually) letters or other symbols representing unknown numbers or values
vertex
for a quadratic function, the lowest (minimum) or the highest (maximum) point on the graph
vertex form
y= a(x – h)^{2} + k, where a, h, and k are constants and (h,k) is the vertex of the parabola
vertical asymptote
a line that a graph approaches more and more closely
vertical shift
movement of a graph up or down
A function f(x) can be moved “k” units up/down by performing the operation f(x) ± k.
x-intercept
the point(s) where the function crosses the x - axis
y-intercept
the point(s) where the function crosses the y - axis
Zero Product Property
If ab = 0, then a = 0 or b = 0 Example: If (x + 1)(x + 2) = 0, then (x + 1) = 0 or (x + 2) = 0.
zeros
the solution(s) of a quadratic equation; a value of x that makes a function f(x) equal to 0, a zero of a function may be real or imaginary