Algebra 1 – Glossary | Open Up HS Math (CCSS), Family

A–F

absolute value

A number’s distance from zero on the number line.

The symbol $| a |$ means the absolute value of $a$ .

Recall that distance is always positive.

The diagram shows that $| 2 | = + 2$ and $| - 2 | = + 2$ .

absolute value function

A function that contains an algebraic expression within absolute value symbols. The absolute value parent function, written as:

$f (x) = | x | = x, if x > 0$

$f (x) = | x | = 0, if x = 0$

$f (x) = | x | = - x, if x < 0$

angle

Two rays that share a common endpoint called the vertex of the angle.

arithmetic mean

The arithmetic mean is also known as the average. The arithmetic mean between two numbers will be the number that is the same distance from each of the numbers. It is found by adding the two numbers and dividing by $2$ .

The arithmetic mean of several numbers is found by adding all of the numbers together and dividing by the number of items in the set:

Example: Find the arithmetic mean of ${5, 6, 7, 8} \Rightarrow \frac{5 + 6 + 7 + 8}{4} = \frac{26}{4} = 6.5$

arithmetic sequence

The list of numbers $3 10 17 24 31$ represents an arithmetic sequence because, beginning with the first term, $3$ , the number $7$ has been added to get the next term. The next term in the sequence will be ( $31 + 7$ ) or $38$ .

The number being added each time is called the constant difference ( $d$ ).

The sequence can be represented by a recursive equation.

In words:

Name the $first term$ . $Next = Previous + d$

Using function notation:

An arithmetic sequence can also be represented with an explicit equation, often in the form $f (n) = d (n - 1) + f (1)$ where $d$ is the constant difference $(d)$ and $f (1)$ is the value of the first term.

The graph of the terms in an arithmetic sequence are arranged in a line.

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

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:

$3 x + 1 y = 14$

$5 x - 3 y = 0$

Here is the augmented matrix for this system:

average rate of change

See rate of change.

bimodal distribution

A bimodal distribution has two main peaks.

The data has two modes.

See also: modes.

binomial

A polynomial with two terms.

bivariate data

Deals with two variables that can change and are compared to find relationships. If one variable is influencing another variable, then you will have bivariate data that has an independent and a dependent variable (ordered pairs). This is because one variable depends on the other for change.

box and whisker plot (box plot)

A one-dimensional graph of numerical data based on the five-number summary, which includes the minimum value, the $25 th$ percentile $(Q_{1})$ , the median, the $75 th$ percentile $(Q_{3})$ , and the maximum value. These five descriptive statistics divide the data into four parts; each part contains $25 %$ of the data.

Boxplots can be vertical or horizontal.

categorical data or categorical variables

Data that can be organized into groups or categories based on certain characteristics, behavior, or outcomes. Also known as qualitative data.

causation

Tells you that a change in the value of the $x$ variable will cause a change in the value of the $y$ variable.

center (statistics)

A value that attempts to describe a set of data by identifying the central position of the data set (as representative of a “typical” value in the set). Measure of center refers to a measure of central tendency (mean, median, or mode).

change factor (pattern of growth)

A change factor is a multiplier that makes each dependent variable grow (or sometimes decrease) as the independent variable increases. Sometimes called the growth factor.

In a geometric sequence it is the common ratio.

In an exponential function it is the base of the exponent.

common ratio (r) (constant ratio)

The change factor or pattern of growth ( $r$ ) in a geometric sequence. To find it divide any output by the previous output.

Example: $f (n)$ is a geometric sequence.

Output $f (n)$	$37$	$74$	$148$	$296$
Input $n$	$1$	$2$	$3$	$4$

Output

$f (n)$

$37$

$74$

$148$

$296$

Input

$n$

$1$

$2$

$3$

$4$

The common ratio is

$r = \frac{output}{previous output} = \frac{296}{148} = 2 or \frac{148}{74} = 2$

commutative property of addition or multiplication

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

completing the square

Completing the Square changes the form of a quadratic function from standard form to vertex form. It can be used for solving a quadratic equation and is one method for deriving the quadratic formula.

$\begin{matrix} standard form & vertex form \\ a x^{2} + b x + c = 0 \Rightarrow & a {(x + d)}^{2} + e \end{matrix}$

$if a = 1,$

$x^{2} + b x + c = 0 \Rightarrow {(x + \frac{b}{2})}^{2} + [c - {(\frac{b}{2})}^{2}]$

$vertex (- \frac{b}{2}, [c - {(\frac{b}{2})}^{2}])$

$Example: \frac{6}{2} = 3 5 - 3^{2}$

$x^{2} + 6 x + 5 = 0 \Rightarrow {(x + 3)}^{2} + (- 4)$

$vertex (- 3, - 4)$

compound inequality in one variable

A compound inequality contains at least two inequalities that are separated by either “and” or “or.”

an inequality that combines two inequalities either so that a solution must meet both conditions (and $\cap$ ) or that a solution must meet either condition (or $\cup$ ).

Examples:

$(- 3, 2]$ can be written as ${x | - 3 < x \leq 2}$

(The value of $x$ in this set must meet both conditions, $x > - 3$ and $x \leq 2$ , which is the same as $x > - 3 \cap x \leq 2$ )

$(- \infty, - 3) \cup [2, \infty)$ can be written as