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A–F

amplitude

The height from the midline (center line) to the maximum (peak) of a periodic graph. Half the distance from the minimum to the maximum values of the range.

For functions of the form or , the amplitude is .

a trigonometric graph with labels for amplitude, midline and distance from minimum to maximumxyamplitudemidlinedistance fromminimum to maximum
angle of rotation in standard position

To represent an angle of rotation in standard position, place its vertex at the origin, the initial ray oriented along the positive -axis, and its terminal ray rotated degrees counterclockwise around the origin when is positive and clockwise when is negative. Let the ordered pair represent the point where the terminal ray intersects the circle.

2 diagrams of circles with terminal rays. The first circle show a positive rotation, and the second shows a negative rotationinitial ray+positive rotationinitial ray-negativerotation
angular speed

Angular speed is the rate at which an object changes its angle in a given time period. It can be measured in . Typically measured in .

arc length

The distance along the arc of a circle. Part of the circumference.

Equation for finding arc length:

Where is the radius and is the central angle in radians.

A circle with a segment created from 2 radii
argument of a logarithm

See logarithmic function.

associative property of addition or multiplication

See properties of operations for numbers in the rational, real, or complex number systems.

asymptote

A line that a graph approaches, but does not reach. A graph will never touch a vertical asymptote, but it might cross a horizontal or an oblique (also called slant) asymptote.

Horizontal and oblique asymptotes indicate the general behavior of the ends of a graph in both positive and negative directions. If a rational function has a horizontal asymptote, it will not have an oblique asymptote.

Oblique asymptotes only occur when the numerator of has a degree that is one higher than the degree of the denominator.

a diagram showing vertical asymptotes between curvesverticalasymptoteverticalasymptote
a diagram showing the oblique asymptote within a 1/x functionobliqueasymptote
a diagram showing the horizontal asymptote within a 1/x functionhorizontal asymptote
augmented matrix

An augmented matrix for a system of equations is a matrix of numbers in which each row represents the constants from one equation (both the coefficients and the constant on the other side of the equal sign) and each column represents all the coefficients for a single variable.

Given the system:

Here is the augmented matrix for this system:

an augmented matrix
base of a logarithm

See logarithmic function (logarithm).

bimodal distribution

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

a bimodal histogram2224446662020204040406060608080800002 modesbimodal distribution
binomial

A polynomial with two terms.

a binomial of (ax b)termtermaddition or subtraction
binomial expansion

When a binomial with an exponent is multiplied out into expanded form.

Example:

Pascal’s triangle (shown) can be used to find the coefficients in a binomial expansion. Each row gives the coefficients to , starting with . To find the binomial coefficients for , use the row and always start with the beginning variable raised to the power of . The exponents in each term will always add up to . The binomial coefficients for are , , , , , and — in that order or

The first 6 rows of Pascal's triangle
Central Limit Theorem (CLT)

This theorem gives you the ability to measure how much your sample mean will vary, without having to take any other sample means to compare it with.

The basic idea of the CLT is that with a large enough sample, the distribution of the sample statistic, either mean or proportion, will become approximately normal, and the center of the distribution will be the true parameter.

clockwise / counterclockwise

clockwise: Moving in the same direction, as the hands on a clock move.

counterclockwise: Moving in the opposite direction, as the hands on a clock move.

a clock with labels for counterclockwise and clockwise directions
closure

A set is closed (under an operation) if and only if the operation on any two elements of the set produces another element of the same set.

A diagram showing that 5 2=7 is closed under addition and 2-5=-3 is not closed under subtraction5 and 2 and 7 are natural numbersThe natural numbers areclosed under addition2 and 5 are natural numbersThe natural numbers are NOTclosed under subtraction-3 isNOT anaturalnumber.
cluster sample

See sample.

common logarithm

A logarithm with base , written , which is shorthand for .

a diagram showing a base 10 logarithmWhen the base is missing, it's 10.A base 10 logarithm is so“common,” it's not written.
commutative property of addition or multiplication

See properties of operations for numbers in the rational, real, or complex number systems.

complex conjugates

A pair of complex numbers whose product is a nonzero real number.

The complex numbers and form a conjugate pair.

The product , a real number.

The conjugate of a complex number is the complex number .

The conjugate of a complex number is represented with the notation .

complex number

A number with a real part and an imaginary part. A complex number can be written in the form , where and are real numbers and is the imaginary unit.

When , the complex number can be written simply as It is then referred to as a pure imaginary number.

the complex number defined as a bi with the square root of negative 1=ithe “a” and “b” are real.imaginary
complex plane

A coordinate plane used for graphing complex numbers, where the horizontal axis is the real axis and the vertical axis is the imaginary axis.

The diagram shows the complex numbers

, ,, and graphed in the complex plane.

a complex plane with the points -1 1i, 2 2i, -2-1i, and 1-2i graphedreal axis–2–2–2–1–1–1111222imaginary axis–2i–2i–2i–1i–1i–1i1i1i1i2i2i2i000
composition of functions

The process of using the output of one function as the input of another function.

Replace with .

a digram showing g(x) as the input for the composition of functions f(g(x))
conjugate pair

A pair of numbers whose product is a nonzero rational number.

The numbers and form a conjugate pair.

The product of , a rational number.

continuous compound interest

Continuously compounded interest means that the account constantly earns interest on the amount of money in the account at any time, which includes the principal and the interest earned previously.

a diagram with the formula for compound interestamountrate of interesttime in yearsthe mathematicalconstant eprincipal–the initialinvestment
control group

The control group is used in an experiment as a way to ensure that your experiment actually works. It is a baseline group that receives no treatment or a neutral treatment. To assess treatment effects, the experimenter compares results in the treatment group to results in the control group.

convenience sample

See sample.

coterminal angles

Two angles in standard position that share the same terminal ray but have different angles of rotation.

The diagram shows a positive rotation () of ray from through to . The dotted arc () shows a negative rotation of ray from through to .

The two angles are coterminal.

a circle diagram showing a shared terminal and coterminal ray with the angle of rotation for each.initial ray
cubic function

a polynomial of degree . The parent function is .

decomposition of functions

Undoing a composite function in terms of its component parts.

The decomposition of f(g(x))=3sin x-1 where f(x)=3x-1 and g(x)=sin x
degree of a polynomial

The power of the term that has the greatest exponent.

The degree of the polynomial 5x^3 8x^2-9x 11 is 3.exponentThe degree is 3.
determinant of a matrix

The determinant of a matrix is a number that is specially defined only for square matrices. If the determinant is not equal to zero, then the matrix has a multiplicative inverse.

For a matrix the determinant can be found using the following rule: (note: the vertical lines, rather than the square brackets, which are used to indicate that we are finding the determinant of the matrix)

distribution curve

A graph of the frequencies of different values of a variable in a statistical distribution.

distributive property of multiplication over addition

The distributive property of multiplication over addition says it’s okay to add within the parentheses first, and then multiply.

Or it’s okay to multiply each term first and then add. The answer works out to be the same.

a diagram labeling the operation in 5(3 9)=(5 times 3) (5 times 9)=14 45=60

The distributive property makes it possible to simplify expressions that include variables. It also makes it possible to factor expressions.

See also properties of operations.

dividend

See division.

division
a diagram of long division

With polynomials:

long division with functions
division algorithm for polynomials

If and are polynomials such that the degree of the degree of , there exists unique polynomials and such that

the labeled diagram for f(x) over d(x)=q(x) r(x) over d(x)dividenddivisorquotientremainderdivisor

where the degree of the degree of .

If , then divides evenly into , making a factor of .

divisor

See division.

domain

The set of all possible -values which will make the function work and will output real -values. A continuous domain means that all real values of included in an interval can be used in the function.

Choosing a smaller domain for a function is called restricting the domain. The domain may be restricted to make the function invertible.

Sometimes the context will restrict a domain.

Other terms that refer to the domain are input values and independent variable.

double angle identities

See trigonometric identities.

elapsed time

The time that has passed since the position of the rider was at the farthest right position on the wheel (standard position with initial ray along the positive -axis).

end behavior

The behavior of a function for -values that are very large (approaching ) and very small (approaching ).

even function

See function: even.

experiment

In an experiment, researchers separate the participants into a control group and a treatment group, and manipulate the variables to try to determine cause and effect. One of the key components of an experiment is that individuals are assigned to treatments and the results are compared.

explicit equation

Relates an input to an output.

Example: ; is the input and is the output. 

The explicit equation is also called a function rule, an explicit formula, or explicit rule.

extraneous solution

A derived solution to an equation that is invalid in the original equation.

factor

Factor (verb): To factor a number means to break it up into numbers that can be multiplied together to get the original number.

Example: Factor : , or , or

Factor (noun): a whole number that divides exactly into another number. In the example above , , , and are all factors of

In algebra factoring can get more complicated. Instead of factoring a number like , you may be asked to factor an expression like .

The numbers and and the variables and are all factors. The variable is a factor that occurs twice.

factor of a polynomial

is a factor of the polynomial function if dividing by leaves no remainder.

frequency

The number of times the event occurred in an experiment or study.

frequency distribution curve, frequency polygon

A frequency distribution curve “smooths out the bumps” in a frequency distribution with a theoretical curve that shows how often an experiment will produce a particular result.

a normal histogram with a frequency distribution curve Frequency distribution curve
function: even, odd

A function is classified as an odd function if . Example: is an example of an odd function. The graph of an odd function is symmetric with respect to the origin. This means it can be rotated and still look the same.

A face saying "What an ODD function! The graph has been rotated 180°, and it looks the same."What an ODD function!The graph has been rotated180°, and it looks the same.
the graph of x^3 and the same graph and grid rotated 180°

A function is classified as an even function if . Example: is an even function. The graph of an even function reflects across the -axis.

A face saying "This graph shows an EVEN function because the value of y is the same when x is positive and when x is negative." along with the graph of y^2–25–25–25–20–20–20–15–15–15–10–10–10–5–5–5555555101010151515202020000y-axis is line of reflectionThis graph shows an EVEN function because the value of y is the same when x is positive and when x is negative.
Fundamental Theorem of Algebra

An degree polynomial function has roots, but some of those roots might be complex numbers.

diagrams showing the the the degree of a polynomial and the roots of that polynomial are the same

G–L

horizontal asymptote

A line that the graph approaches but does not reach. Exponential functions have a horizontal asymptote. The horizontal asymptote is the value the function approaches as either gets infinitely larger or smaller. An asymptote is an imaginary line, but it is often shown as a dotted line on the graph.

As gets smaller, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^x and its horizontal asymptote of y=0x–10–10–10–5–5–5555y–5–5–5555101010000

As gets larger, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^-x and its horizontal asymptote of y=0x–5–5–5555101010y–5–5–5555101010000

As gets smaller, the graph of approaches the horizontal asymptote, .

the graphs of f(x)=2^x-3 and its horizontal asymptote of y=-3x–10–10–10–5–5–5555y–5–5–5555101010000

See also asymptote.

horizontal shift

See transformations on a function.

identities (additive/multiplicative) with matrices
diagrams showing the additive and multiplicative identity properties of matricesAdditive IdentityMultiplicative IdentityAdditive InverseMultiplicative Inverse
diagrams showing the additive and multiplicative inverse properties of matricesAdditive IdentityMultiplicative IdentityAdditive InverseMultiplicative Inverse
identity: additive, multiplicative
Two faces thinking about the additive and multiplicative identity properties of 5 0=5 and 5 times 1 =5What number can Iadd to a number toget the same numberfor the answer?What can I multiply anumber by to get thesame number for theanswer?Zero is theadditive identity.One is themultiplicative identity.

See also Properties of Operations.

imaginary number

See complex number.

independent variable / dependent variable

In a function, the independent variable is the input to the function rule and the dependent variable is the output after the function rule has been applied. Also called ordered pairs, coordinate pairs, input-output pairs. The domain describes the independent variables and the range describes the dependent variables.

diagram showing showing the independent and dependent variables in the function f(x)=5(x)-7
inference (statistics)

The use of results from a sample to draw conclusions about a population.

initial ray

See angle of rotation in standard position.

input-output pair

Input and output pairs are related by a function rule. Also called ordered pairs, coordinate pairs, independent and dependent variables. If is an input-output pair for the function , then is the input, is the output and .

A diagram representing an input/output pair for f(x)=5x-7; x=3
interval of increase or decrease

In an interval of increase, the -values are increasing. In an interval of decrease, the -values are decreasing. When describing an interval of increase or decrease, the -values that correspond to the increasing or decreasing -values are named.

a diagram showing increasing, decreasing and constant intervalsx–2–2–2222444666y–8–8–8–6–6–6–4–4–4–2–2–2222444000
interval of plausible values

A range of likely values for the population parameter, based on a sample statistic.

inverse cosine function
A diagram showing the inverse cosine function–18–18–18–17–17–17–16–16–16–15–15–15–14–14–14–13–13–13–12–12–12–11–11–11–10–10–10–9–9–9–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010–7π / 2–7π / 2–7π / 2–3π–3π–3π–5π / 2–5π / 2–5π / 2–2π–2π–2π–3π / 2–3π / 2–3π / 2–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ000kk
inverse function
several diagrams that show different representations for the inverse of f(x) using a table, graph, and equations.
inverse sine function
a diagram showing the inverse sine function–18–18–18–17–17–17–16–16–16–15–15–15–14–14–14–13–13–13–12–12–12–11–11–11–10–10–10–9–9–9–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444555666777888999101010–7π / 2–7π / 2–7π / 2–3π–3π–3π–5π / 2–5π / 2–5π / 2–2π–2π–2π–3π / 2–3π / 2–3π / 2–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2000kk
inverse tangent function
a diagram showing the inverse tangent function–12–12–12–10–10–10–8–8–8–6–6–6–4–4–4–2–2–2222444666888101010121212–3π / 2–3π / 2–3π / 2–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ3π / 23π / 23π / 22π2π2π5π / 25π / 25π / 2000fff
inverse: additive, multiplicative

The number you add to a number to get zero is the additive inverse of that number. Every nonzero real number has a unique additive inverse. Zero is its own additive inverse. . For every there exists so that

The reciprocal of a nonzero number is the multiplicative inverse of that number. The reciprocal of is because . The product of a real number and its multiplicative inverse is . Every real number has a unique multiplicative inverse.

invertible function

A function is invertible if and only if its inverse is defined and is a function.

graphs comparing the function of x^3 and x^2 and their inverse graphs

If a function is not invertible across its entire domain, the domain can be restricted so that it is invertible.

See one-to-one function.

irrational number

An irrational number is a real number that cannot be written in the form , where and are integers and . is often used as the symbol for irrational. The bar on top means NOT rational.

the classification of all number systems
leading coefficient

The number written in front of the variable with the largest exponent.

a diagram showing the polynomial 2x^5 7x^2-13 has coefficients of 2, 7, and -13 with a leading coefficient of 2
linear function
several diagrams modeling linear functions, including tables and graphs. Equations for linear functions are defined as y=mx b, y=m(x-x1) y1, and Ax By=Clinear functionslope-intercept formm = slopeb = y-interceptpoint-slope formyou need slope and a point:standard formdomain: all real numbersrange: all real numbersunless restricted. unless restricted.graph is a linerate of change (slope) is constant1st difference is constantThe function can increase,decrease, or remain constant. 2122
logarithmic function (logarithm)

The inverse of an exponential function is called a logarithmic function.

If , then .

The base of the log and the base of the exponent match. A logarithm has 3 parts: the argument, the base, and the answer.

a diagram showing the parts of a logarithmic function.

M–R

margin of error

The margin of error is a statistic expressing the amount of random sampling error in the results of a survey. The larger the margin of error, the less confidence one should have that a poll result will reflect the result of a survey of the entire population.

matrix (properties of operations)

Associative Property of Addition

Examples with Real Numbers

Examples with Matrices

Associative Property of Multiplication

Examples with Real Numbers

Examples with Matrices

Commutative Property of Addition

Examples with Real Numbers

Examples with Matrices

Commutative Property of Multiplication

Examples with Real Numbers

Examples with Matrices

Distributive Property of Multiplication Over Addition

Examples with Real Numbers

Examples with Matrices

matrix row reduction

Row reduction is the process of using row operations to reduce a matrix to row reduced form. This procedure is used to solve systems of linear equations, invert matrices, compute determinants, and do many other things.

To row reduce a matrix:

  • Replace an equation (or a row of the matrix) with a multiple of that equation (or row); Example: .

  • Replace an equation (or a row of the matrix) with the sum or difference of that equation (or row) with a multiple of another equation (or row); Example:

maximum / minimum

Maximum is the point at which a function’s value is greatest.

Minimum is the point at which a function’s value is the least.

A cubic function with points showing the maximum and minimum.–2–2–2–1–1–1111222–1–1–1111000maximumminimum
midline of a trigonometric function

A horizontal axis that is used as the reference line about which the graph of a periodic function oscillates. The equation of the midline is , where is the vertical translation of the function.

the diagram of the sine function with the amplitude, midline, and distance from maximum to minimum labeledxyamplitudemidline
mode(s)

The measure of central tendency for a one-variable data set that is the value(s) that occurs most often.

Types of modes include: uniform (evenly spread- no obvious mode), unimodal (one main peak), bimodal (two main peaks), or multimodal (multiple locations where the data is relatively higher than others).

a histogram with a uniform distribution111222333444555202020404040606060808080000uniform distribution
a histogram with a unimodal distribution222444666888202020404040606060808080100100100000unimodal distributionone mode
a histogram with a bimodal distribution111222333444555202020404040606060808080000bimodal distribution2 modes
a histogram with a multimodal distribution222444666888202020404040606060808080100100100000multimodal distributionmany modes

See measures of central tendency.

modulus

The modulus of the complex number is This is the distance between the origin and the point in the complex plane.

For two points in the complex plane, the distance between the points is the modulus of the difference of the two complex numbers. The formula looks a lot like the formula for finding the distance between two points.

Example: Given two complex numbers:

and , the distance between them is

Find the distance between and .

a imaginary coordinate plane with the points (1 3i), and (-2,-1i) graphedx–2–2–2–1–1–1111y–1–1–1111222333000
multiplicity

The multiplicity of each zero is the number of times that its corresponding factor appears. If , the zeros or roots of are multiplicity and multiplicity .

The multiplicity of a root affects the shape of the graph of a polynomial. If a root of a polynomial has odd multiplicity, the graph will cross the -axis at the root.

Graph of

multiplicity and multiplicity

the graph of p(x)=(x 2)(x-1)(x-1)(x-1) with the x intercepts of -2 and 1 with an odd multiplicity labeledx–4–4–4–2–2–2222y–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222000odd multiplicity
the graph of q(x)=(x-2)(x-1)(x-1) with the x intercepts of -2 and 1 with an even multiplicity labeledx–2–2–2222y–2–2–2222444000even multiplicity

If a root of a polynomial has even multiplicity, the graph will touch the -axis at the root but will not cross the -axis.

Graph of

multiplicity and multiplicity

natural logarithm

A logarithm with base , written , which is shorthand for .

normal distribution

An arrangement of a data set in which most values cluster in the middle of the range and the rest taper off symmetrically toward either extreme. Features of a normal distribution include:

  1. A normal distribution is symmetric.

  2. The mean, median, and mode are equal in a normal distribution.

  3. The frequency curve of a normal distribution is symmetric.

  4. A normal curve has points of inflection at standard deviation from the mean.

  5. A normal distribution has a single mode.

of the distribution will be standard deviations from the mean. of the distribution will be within standard deviations from the mean. of the distribution will be within standard deviations from the mean.

a normal distribution curve
observational study

a study in which the researcher simply observes the subjects without interfering.

In this type of study, researchers observe the behavior of the participants/subjects without trying to influence it in any way so they can learn about the parameter of interest.

odd function

See function: odd.

one-to-one function

A function is said to be one-to-one if no two elements of the domain of correspond to the same element in the range of . If no horizontal line intersects the graph of the function in more than one point, the function is one-to-one.

The function is a one-to-one function.

It is an invertible function.

The graph of x^3 with a dotted line of y=2–3–3–3–2–2–2–1–1–1111222333–3–3–3–2–2–2–1–1–1111222333000

The function is NOT a one-to-one function because each -value occurs twice for different elements of the domain, except at the vertex. It is not an invertible function.

the graph of x^2 with a dotted line of y=1–2–2–2–1–1–1111222–1–1–1111222000
parameter

A number that describes a characteristic of a population (such as the mean or the standard deviation).

parameter of interest (statistics)

A parameter of interest is what your data is focused on.

The thing we want to know about the population.

A number, such as the mean or standard deviation, that describes an entire population. Any numerical quantity that characterizes a given population or some aspect of it. This means the parameter tells us something about the whole population.

parent function

The most basic form of a function. A parent function can be transformed to create a family of functions.

Pascal’s triangle

An array of numbers forming a triangle named after a famous mathematician Blaise Pascal. The top number of the triangle is , as well as all the numbers on the outer sides. To get any term in the triangle, you find the sum of the two numbers above it. The top number is considered row of the triangle.

The first 6 rows of Pascal's triangle
period of a cyclical function

The time it takes for one complete cycle of a cyclical motion to occur. The diagram shows the graph of . The graph begins at . At the graph begins to repeat because it has completed one cycle. The period is .

the graph of sine of xxπππ2π2π2π3π3π3π4π4π4πy000
period of rotation

The period of rotation is the amount of time for one complete rotation of the Ferris wheel.

phase shift

For trigonometric functions, a horizontal transformation of a graph is referred to as a phase shift.

point of inflection

A point on a curve where the curve changes from being concave down to concave up or vice versa.

the graph of a cubic function with labels that show where the graph is concave downward, concave upward, and the inflection pointx–3–3–3–2–2–2–1–1–1111222y–1–1–1111222333444000concave downwardconcave upwardinflection point

In the normal curve it is one standard deviation away in either direction from the mean.

a normal distribution curve from mu-3 sigma to mu 3 sigma
polar coordinates

A method of representing points in a plane with ordered pairs in the form where is the distance of the point from the origin and is the angle of rotation of the point from the positive -axis.

polynomial function

A function of the form:

where all of the exponents are positive integers and all of the coefficients are constants.

population (in statistics)

The group of individuals you want to study in order to answer your research question

population mean

The population mean is an average of a group characteristic or item of interest.

The symbol ‘' represents the population mean.

population parameter

A population parameter is the actual value of a statistical measure such as the mean or standard deviation for a given population.

population proportion

A population proportion is a fraction of the population that has a certain characteristic. The letter is used for the population proportion. It can be written as a fraction e.g. or as a decimal .

product

The result of multiplication is a product.

a diagram showing (x a)(x b)=x^2 ax bx b^2
properties of operations for numbers in the rational, real, or complex number systems

The letters , , and stand for arbitrary numbers in the rational, real, or complex number systems. The properties of equality are true in these number systems.

Associative property of addition

Commutative Property of addition

Additive identity property of

Existence of additive inverses

For every , there exists , so that .

Associative property of multiplication

Commutative Property of multiplication

Multiplicative identity property of

Existence of multiplicative inverses

For every , there exists , so that .

Distributive property of multiplication over addition

quadrantal angle

An angle in standard position with terminal side on the -axis or -axis. Some examples are the angles located at , , , , .

A blank graph with each axis labeled as 0°, 90°, 180°, 270°, and 360°y
quadratic equations

An equation that can be written in the form

Standard form:

Example:

Factored form:

Vertex form:

Recursive form:

(Note: Recursive forms are only used when the function is discrete.)

quadratic function
Several diagrams representing a quadratic function, including an area model, growing steps, a 2nd difference table, and parabolic graphs.quadratic functionvertex formfactored formstandard formgraph is a parabola
quadratic inequality

A function whose degree is and where the is not always exactly equal to the function. These types of functions use symbols called inequality symbols that include the symbols we know as less than , greater than , less than or equal to , and greater than or equal to .

Example:

quotient

See division.

radian

A unit of measure for angles. One radian is the angle made at the center of a circle by an arc whose length is equal to the radius of the circle.

The ratio of the length of an intercepted arc to the radius of the circle on which that arc lies.

A circle with the radius labeled and an intercepted arc that has the same length as the radius
range of a function

All the resulting -values obtained after substituting all the possible -values into a function. All of the possible outputs of a function. The values in the range are also called dependent variables.

rate of change (slope)

A rate that describes how the output of a function changes in relation to the input.

Functions are defined by their rates of change.

In a linear function, if is the independent variable and is the dependent variable, the rate of change equals and is called the slope.

An exponential function has an exponential rate of change.

A quadratic function has a linear rate of change.

A cubic has a quadratic rate of change.

rational function

A function is called a rational function if and only if it can be written in the form where and are polynomials in and is not the zero polynomial.

reciprocal trigonometric functions

The reciprocal of the sine, cosine, and tangent ratios.

a right triangle with the vertices labeled A, B, and C and each opposite side labeled a, b, and c.
rectangular coordinate system

Also called the Cartesian coordinate system, it’s the two-dimensional plane that allows us to see the shape of a function by graphing.

Each point in the plane is defined by an ordered pair. Order matters! The first number is always the -coordinate; the second is the -coordinate.

The coordinate plane with all quadrants labeled and the points A(2,3), B(-2,2), D(2,-1), and E (-2,-3)x–12–12–12–11–11–11–10–10–10–9–9–9–8–8–8–7–7–7–6–6–6–5–5–5–4–4–4–3–3–3–2–2–2–1–1–1111222333444y–4–4–4–3–3–3–2–2–2–1–1–1111222333444000AAABBBCCCDDDQ IIQ IQ IVQIII
recursive equation

Also called recursive formula or recursive rule. See examples under arithmetic sequence and geometric sequence, and quadratic equations.

reference triangle

A right triangle that is drawn connecting the terminal ray of an angle in standard position to the -axis. In the diagram, is the reference triangle.

A right triangle in Quadrant II made with the x axis and a terminal ray and an angle of rotationreference triangleinitial ray
reflection

A reflection is a rigid transformation (isometry). In a reflection, the pre-image and image points are the same distance from the line of reflection; the segment connecting corresponding points is perpendicular to the line of reflection.

The orientation of the image is reversed.

a reflection of a polygon over a line
remainder

See division.

remainder theorem for polynomials

When a polynomial f is divided by , the remainder equals .

Why is this true? The division algorithm can be used to prove the remainder theorem.

a diagram showing the remainder theorem for polynomials
restricted domain

Limiting the domain of a function so that its inverse is also a function.

roots: real and imaginary

The solutions of an equation in the form .

row reductions of matrices

To solve a system using row reduction of matrices:

  • Perform elementary row operations to get a 1 in one of the columns.

  • Get zeros in all of the other rows for that column by adding a constant multiple of the row to each other row.

  • Perform elementary row operations to get a 1 in another column.

  • Create zeros in all of the other rows for that column by adding a constant multiple of the row to each other row.

  • Continue this process until each column contains a 1 and there are 0’s everywhere else, except in the augmented column that will contain the solutions to the system.

S–X

sample

A part of a population selected to represent the entire population. Sampling is the process of selecting and studying a sample from a population in order to make conjectures about the entire population. A good sample represents the target population.

Types of samples:

simple random sample - one in which every possible sample (of the same size) has an equal chance of being selected from the target population.

systematic sample – A method of choosing a random sample from among a larger population. The process of systematic sampling typically involves first selecting a fixed starting point in the larger population and then obtaining subsequent observations by using a constant interval between samples taken.

cluster sample – With cluster sampling, the researcher divides the population into separate groups, called clusters. Then, a simple random sample of clusters is selected from the population. The researcher conducts his analysis on data from the sampled clusters.

stratified random sample – With stratified sampling, the researcher divides the population into separate groups, called strata. Then, a probability sample (often a simple random sample ) is drawn from each group.

convenience sample – made up of people who are easy to reach.

volunteer sample - made up of people who self-select into the survey.

sample mean

The sample mean is simply the average of all the measurements in the sample. If the sample is random, then the sample mean can be used to estimate the population mean. The symbol for sample mean is ( bar)

sample proportion

the proportion of people from the sample who fell into a certain group

The symbol is ( - hat).

sample statistic

A statistic or sample statistic is any quantity computed from values in a sample that is used for a statistical purpose. It’s a piece of information you get from a fraction of a population.

simple random sample

See sample.

simulation

A model of random events, often using technology

One purpose is to test a hypothesis without doing an actual experiment. They are sometimes used because they are cheaper, faster, and less risky than an actual study.

skewed distribution

When most data is to one side leaving the other with a ‘tail’. Data is skewed to side of tail. (if tail is on right side of data, then it is skewed right).

a histogram with a distribution that is skewed rightxskewed rightmodemedianmean40608020123456780
a histogram with a distribution that is skewed left111222333444555666777888202020404040606060808080000skewed leftmodemedianmean
slant asymptote

See asymptote.

slope

A linear function has a constant slope or rate of change. You can count the slope of a line on a graph by counting how much it changes vertically each time you move one unit horizontally. A move down is negative and a move to the left is negative.

If you know two points on the graph, you can use the slope formula. Given two different points and

is the symbol for slope.

a straight line going through the point (0,-1) that has labels along the line of up 2 and right 1. –3–3–3–2–2–2–1–1–1111222333–3–3–3–2–2–2–1–1–1111222333000
special right triangles

There are two special right triangles. They are special because they can be solved without using trigonometry.

two 45°-45°-90°right triangles with their sides labeled45°45°45°45°If hypotenuse is known (x), then side is45 - 45 right triangleIf side lengths are known (x),then the hypotenuse is
three 30°-60°-90°right triangles with their sides labeled60°30°30°60°30°60°If side opposite 60° is known (x), then hypotenuse isand side opposite 30° isand the side opposite 30° isIf hypotensue is known (x),then side opposite of 60° is
standard deviation

A number used to tell how measurements for a group are spread out from the average (mean), or expected value. A low standard deviation means that most of the numbers are close to the average. A high standard deviation means that the numbers are more spread out. Symbol for standard deviation . (sigma)

standard normal distribution

The standard normal distribution is a normal distribution with a mean of zero and standard deviation of 1. The standard normal distribution is centered at zero and the degree to which a given measurement deviates from the mean is given by the standard deviation.

* normal distributions do not necessarily have the same means and standard deviations

stratified random sample

See sample.

subtraction of polynomials

Subtraction and addition are opposite operations. This is true with polynomials. The diagram shows how the parts of an addition problem and a subtraction problem are related.

a diagram showing the addition and subtraction of polynomialsaddendssumminuendsubtrahenddifference
subtrahend

See subtraction of polynomials.

sum and difference identities

See trigonometric identities.

symbols for sample statistics and corresponding population parameters

Sample Statistic

Population Parameter

Description

number of members of sample or population

“-bar”

“mu”

mean

“sigma”

standard deviation

“rho”

coefficient of linear correlation

“-hat”

proportion

systematic sample

See sample.

terminal ray or side

The side of an angle in standard position that is not on the positive -axis but has an endpoint at the origin or center of rotation.

a circle with the terminal ray, and initial ray labeled
transformations on a function (rigid)

A shift up, down, left, or right, or a vertical or horizontal reflection on the graph of a function is called a rigid transformation.

Vertical shift

Up when

Down when

The vertical shift of a parabolax–4–4–4–3–3–3–2–2–2–1–1–1111y–1–1–1111222333000

Horizontal shift

Left when

Right when

the horizontal shift of a parabolax–3–3–3–2–2–2–1–1–1111222333444y–2–2–2–1–1–1111222333000

Reflection

reflection over the -axis

The reflection of a parabola over the x axisx–1–1–1111y–1–1–1111000

reflection over the -axis

the reflection of a cubic function over the y axisx–1–1–1111y–1–1–1111000

A dilation is a nonrigid transformation. It will make the function changes faster or slower depending on the value of . If , it will grow faster and look like it has been stretched. If , the function will grow more slowly and will appear to be fatter.

treatment group

The treatment group consists of participants who receive the experimental treatment whose effect is being studied.

The control group consists of participants who do not receive the experimental treatment being studied.

trigonometric functions

Right triangle trigonometry can be extended to define trigonometric functions for angles of rotation of any value, including negative values. To do so, a standard position for an angle of rotation is defined: locate the vertex of the angle of rotation at the origin of a Cartesian coordinate system, with the initial ray pointed along the positive -axis. A counter-clockwise rotation is considered positive while a clockwise rotation is considered negative. With this new definition of the trigonometric functions, trigonometry can be applied to periodic behavior.

a circle with the terminal ray, and initial ray labeled
trigonometric identities

Statements that are true for all values of (theta).

Tangent and Cotangent Identities

 

Reciprocal Identities

Pythagorean Identities

Even/Odd Formulas

Periodic Formulas

If is an integer,

Double Angle Formulas

Sum and Difference Formulas

If is an angle in degrees and is an angle in radians, then:

Definitions of the Inverse Trigonometric Functions

Domain:

Range:

the graph of the inverse sine function–2–2–2–1–1–1111222–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ000

Domain:

Range:

the graph of the inverse cosine function–2–2–2–1–1–1111222–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ000

Domain:

Range:

the graph of the inverse tangent function–3–3–3–2–2–2–1–1–1111222333–π–π–π–π / 2–π / 2–π / 2π / 2π / 2π / 2πππ000
trinomial

A polynomial with three terms.

a diagram showing ax^2 bx c has 3 terms3 terms
unimodal

See mode(s).

unit circle

A circle with radius of one unit. The equation of a unit circle with center is .

The unit circle is a useful tool when studying trigonometric functions.

  • Radian measure is the ratio . On the unit circle , so the radian measure is the arc length.

  • Sine is the ratio . On the unit circle , so the sine is the -coordinate.

  • Cosine is the ratio . On the unit circle , so the cosine is the -coordinate.

Example: In the unit circle shown, point is defined by the coordinates . Since , is and is . The arc length is or .

the unit circle with a 30-60-90 triangle shown and labeled
vertical asymptote

See asymptote.

vertical height

The perpendicular distance from the ground up to a designated position.

vertical shift

See transformations on a function (rigid).

vertical stretch

See transformations on a function (non-rigid).

volunteer sample

See sample.

x-intercept

The point(s) where a line or a curve cross the -axis. The -value of the point will be . A non-horizontal line will only cross the -axis once. A curve could cross the -axis several times.

a line passing through the points (-5,0) and (0,2)x–6–6–6–4–4–4–2–2–2y222000(-5, 0)(-5, 0)(-5, 0)
a parabola with a vertex at (-1,-4) passing through the points (-3,0) and (1,0)x–4–4–4–2–2–2222y–4–4–4–2–2–2222000(-3, 0)(-3, 0)(-3, 0)(1, 0)(1, 0)(1, 0)

Y–Z

z-score

The number of standard deviations that a given -value lies from the mean in a normal distribution. The formula for transforming a data point from any normal distribution to a standard normal distribution:

zeros (of a function)

The values of the independent variable (x-values) that make the corresponding values of the function (-values) equal to zero. Real zeros correspond to -intercepts of the graph of a function.

zeros, roots, solutions

The real solutions to a quadratic equation are where it is equal to zero. They are also called zeros or roots. Real zeros correspond to the -intercepts of the graph of a function.

a parabola with the vertex A and passing through the points B(10,0) and C(2,0)x222444666888101010121212141414y–6–6–6–4–4–4–2–2–22224440002 and 10 are theof the quadraticzerosrootssolutionsx-intercepts