Math III – Glossary | Open Up HS Math (CCSS), Student

A–F

acute angle

Unit 5 Lesson 6

An angle whose measure is between $0 °$ and $90 °$ .

$∠ A B C$ is an acute angle.

acute triangle

Unit 5 Lesson 6

A triangle with three acute angles.

Angles $A$ , $B$ , and $C$ are all acute angles.

Triangle $A B C$ is 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.

amplitude

Unit 6 Lesson 4

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 $y = A \sin b θ$ or $y = A \cos b θ$ , the amplitude is $| A |$ .

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 $x$ -axis, and its terminal ray rotated $θ$ degrees counterclockwise around the origin when $θ$ is positive and clockwise when $θ$ is negative. Let the ordered pair $(x, y)$ represent the point where the terminal ray intersects the circle.

angular speed

Unit 6 Lesson 2, Unit 6 Lesson 4

Angular speed is the rate at which an object changes its angle in a given time period. It can be measured in $(\frac{degrees}{second})$ . Typically measured in $(\frac{radians}{second})$ .

arc length

Unit 6 Lesson 7

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

Equation for finding arc length:

$s = r θ$

Where $r$ is the radius and $θ$ is the central angle in radians.

argument of a logarithm

Unit 2 Lesson 1

See logarithmic function.

asymptote

Unit 2 Lesson 2, Unit 4 Lesson 1

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 $f (x)$ has a degree that is one higher than the degree of the denominator.

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.

binomial

Unit 3 Lesson 4

A polynomial with two terms.

binomial expansion

Unit 3 Lesson 4

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

Example:

${(a + b)}^{2} = (a + b) (a + b) = 1 a^{2} + 2 a b + 1 b^{2}$

Pascal’s triangle (shown) can be used to find the coefficients in a binomial expansion. Each row gives the coefficients to ${(a + b)}^{n}$ , starting with $n = 0$ . To find the binomial coefficients for ${(a + b)}^{n}$ , use the $n th$ row and always start with the beginning variable raised to the power of $n$ . The exponents in each term will always add up to $n$ . The binomial coefficients for ${(a + b)}^{5}$ are $1$ , $5$ , $10$ , $10$ , $5$ , and $1$ — in that order or $1 a^{5} + 5 a^{4} b + 10 a^{3} b^{2} + 10 a^{2} b^{3} + 5 a b^{4} + 1 b^{5}$

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.

closure

Unit 3 Lesson 6

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.

cluster sample

Unit 9 Lesson 5

See sample.

common logarithm

Unit 2 Lesson 5

A logarithm with base $10$ , written $\log x$ , which is shorthand for $\log_{10} x$ .

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.

$f (x) = 3 x - 1; g (x) = 5 x + 1$

Replace $x$ with $5 x + 1$ .

$f (g (x)) = 3 (5 x + 1) - 1$

$f (g (x)) = 15 x + 3 - 1 = 15 x + 2$

conjugate pair

Unit 3 Lesson 8

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

The numbers $(a + \sqrt{b})$ and $(a - \sqrt{b})$ form a conjugate pair.

The product of $(a + \sqrt{b}) (a - \sqrt{b}) = a^{2} - {(\sqrt{b})}^{2} = a^{2} - b$ , 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.

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 ( $blue$ ) of ray $A C$ from $B$ through $F$ to $C$ . The dotted arc ( $green$ ) shows a negative rotation of ray $A C$ from $B$ through $D$ to $C$ .

The two angles are coterminal.

cross-section of a solid

Unit 5 Lesson 1

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

cubic function

Unit 3 Lesson 1, Unit 3 Lesson 2

a polynomial of degree $3$ . The parent function is $y = x^{3}$ .

decomposition of functions

Unit 8 Lesson 5

Undoing a composite function in terms of its component parts.

degree of a polynomial

Unit 3 Lesson 2

The power of the term that has the greatest exponent.

density

Unit 5 Lesson 4

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.

$density = \frac{mass}{volume}$

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.

$density = \frac{how many}{space or area}$

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.

dividend

Unit 3 Lesson 5

See division.

division

Unit 3 Lesson 5

With polynomials:

division algorithm for polynomials

Unit 3 Lesson 5

If $f (x)$ and $d (x) \neq 0$ are polynomials such that the degree of $d (x) \leq$ the degree of $f (x)$ , there exists unique polynomials $q (x)$ and $r (x)$ such that

where the degree of $r (x) <$ the degree of $d (x)$ .

If $r (x) = 0$ , then $d (x)$ divides evenly into $f (x)$ , making $d (x)$ a factor of $f (x)$ .

divisor

Unit 3 Lesson 5

See division.

domain

Unit 1 Lesson 2

The set of all possible $x$ -values which will make the function work and will output real $y$ -values. A continuous domain means that all real values of $x$ 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 $3$ -D solid.

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

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 $x$ -axis).

end behavior

Unit 3 Lesson 9

The behavior of a function $y = f (x)$ for $x$ -values that are very large (approaching $\infty$ ) and very small (approaching $- \infty$ ).

even function

Unit 3 Lesson 9, Unit 7 Lesson 4

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: $f (t) = 4 t + 1$ ; $t$ is the input and $f (t)$ 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

Unit 3 Lesson 4

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 $75$ : $75 = 3 \times 25$ , or $15 \times 5$ , or $3 \times 5 \times 5$

Factor (noun): a whole number that divides exactly into another number. In the example above $3$ , $5$ , $15$ , and $25$ are all factors of $75.$

In algebra factoring can get more complicated. Instead of factoring a number like $75$ , you may be asked to factor an expression like $15 a^{2} b$ .

The numbers $3$ and $5$ and the variables $a$ and $b$ are all factors. The variable $a$ is a factor that occurs twice.

factor of a polynomial

Unit 3 Lesson 7

$(x - r)$ is a factor of the polynomial function $y = P (x)$ if dividing $P (x)$ by $x - r$ leaves no remainder.

frequency

Unit 7 Lesson 3

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.

frustum

Unit 5 Lesson 3

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.

function: even, odd

Unit 3 Lesson 9

A function $f (x)$ is classified as an odd function if $f (- θ) = - f (θ)$ . Example: $f (x) = x^{3}$ 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 $180 °$ and still look the same.

A function $f (x)$ is classified as an even function if $f (- θ) = f (θ)$ . Example: $f (x) = x^{2}$ is an even function. The graph of an even function reflects across the $y$ -axis.

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 $x$ either gets infinitely larger or smaller. An asymptote is an imaginary line, but it is often shown as a dotted line on the graph.

$f (x) = 2^{x}$

As $x$ gets smaller, the graph of $f (x)$ approaches the horizontal asymptote, $y = 0$ .

$h (x) = 2^{- x}$

As $x$ gets larger, the graph of $h (x)$ approaches the horizontal asymptote, $y = 0$ .

$g (x) = 2^{x} - 3$

As $x$ gets smaller, the graph of $g (x)$ approaches the horizontal asymptote, $y = - 3$ .