A–F

acute angle
Unit 5 Lesson 6

An angle whose measure is between and .

is an acute angle.

an acute angle
acute triangle
Unit 5 Lesson 6

A triangle with three acute angles.

Angles , , and are all acute angles.

Triangle is an acute triangle.

an acute triangle
Ambiguous Case of the Law of Sines
Unit 5 Lesson 8

The Ambiguous Case of the Law of Sines occurs when we are given SSA information about the triangle. Because SSA does not guarantee triangle congruence, there are two possible triangles.

To avoid missing a possible solution for an oblique triangle under these conditions, use the Law of Cosines first to solve for the missing side. Using the quadratic formula to solve for the missing side will make both solutions become apparent.

ambiguous case of the law of sines
ambiguous case of the law of sines

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
Unit 6 Lesson 3

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 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
Unit 6 Lesson 7

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
Unit 2 Lesson 1

See logarithmic function.

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
base of a logarithm
Unit 2 Lesson 1

See logarithmic function (logarithm).

bimodal distribution
Unit 9 Lesson 1

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

a bimodal histogram2224446662020204040406060608080800002 modesbimodal distribution

A polynomial with two terms.

a binomial of (ax b)termtermaddition or subtraction
binomial expansion
Unit 3 Lesson 4

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)
Unit 9 Lesson 8

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
Unit 6 Lesson 2

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

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
Unit 9 Lesson 5

See sample.

common logarithm
Unit 2 Lesson 5

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.
composition of functions
Unit 8 Lesson 2, Unit 8 Lesson 4

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
Unit 3 Lesson 8

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
Unit 2 Lesson 6

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
Unit 9 Lesson 6

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
Unit 9 Lesson 5

See sample.

coterminal angles
Unit 6 Lesson 3

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
cross-section of a solid
Unit 5 Lesson 1

The face formed when a three-dimensional object is sliced by a plane.

cross-section of a solid

a polynomial of degree . The parent function is .

decomposition of functions
Unit 8 Lesson 5

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
Unit 3 Lesson 2

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.

In science, density describes how much space an object or substance takes up (its volume) in relation to the amount of matter in that object or substance (its mass). If an object is heavy and compact, it has a high density. If an object is light and takes up a lot of space, it has a low density.

Density can also refer to how many people are crowded into a small area or how many trees are growing in a small space or a large space. In that sense it is a comparison of compactness to space.

densitymore dense
disc or disk
Unit 5 Lesson 2

See solid of revolution.

distribution curve
Unit 9 Lesson 1

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

See division.

a diagram of long division

With polynomials:

long division with functions
division algorithm for polynomials
Unit 3 Lesson 5

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 .

See division.

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
Unit 7 Lesson 7

See trigonometric identities.

edge / face / vertex of a 3-D solid
Unit 5 Lesson 1

Edge: The line that is the intersection of two planes.

Face: A flat surface on a -D solid.

Vertex: (pl. vertices) Each point where two or more edges meet; a corner.

edge / face / vertex of a 3-D solidfacevertexedge
elapsed time
Unit 6 Lesson 2

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
Unit 3 Lesson 9

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

See function: even.

experiment
Unit 9 Lesson 6

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
Unit 2 Lesson 6

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
Unit 4 Lesson 7

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

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
Unit 3 Lesson 7

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

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

frequency distribution curve, frequency polygon
Unit 9 Lesson 1

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

The part of a solid such as a pyramid or a cone that remains after cutting off a top portion with a plane parallel to the base.

frustumfrustum
function: even, odd
Unit 3 Lesson 9

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.

G–L

horizontal asymptote
Unit 4 Lesson 1

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
Unit 2 Lesson 2

See transformations on a function.

independent variable / dependent variable
Unit 1 Lesson 1

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)
Unit 9 Lesson 8

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

initial ray
Unit 6 Lesson 3

See angle of rotation in standard position.

input-output pair
Unit 1 Lesson 1

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
Unit 1 Lesson 2

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
Unit 9 Lesson 9

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

inverse cosine function
Unit 7 Lesson 8
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
Unit 1 Lesson 1
several diagrams that show different representations for the inverse of f(x) using a table, graph, and equations.
inverse sine function
Unit 7 Lesson 8
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
Unit 7 Lesson 8
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 / 25π / 25π / 25π / 2000fff
invertible function
Unit 1 Lesson 2

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
Unit 2 Lesson 6

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
law of cosines
Unit 5 Lesson 7
law of sines

For any triangle with angles , , and , and sides of lengths , , and , where is opposite , and is opposite and is opposite , these equalities hold true:

The law of cosines is useful for finding:

  • the third side of a triangle when we know two sides and the angle between them.

  • the angles of a triangle when we know all three sides.

law of sines
Unit 5 Lesson 6

For any triangle with angles , , and , and sides of lengths , , and , where is opposite , and is opposite and is opposite , these equalities hold true: .

law of sines
leading coefficient
Unit 3 Lesson 9

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
Unit 1 Lesson 1
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=C