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

A–F

acute angle

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

$∠ A B C$ is an acute 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.

average rate of change

See rate of change.

bimodal distribution

A bimodal distribution has two main peaks.

The data has two modes.

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

Output $f (n)$	$19$	$25$	$31$	$37$
Input $n$	$1$	$2$	$3$	$4$

G–L

geometric sequence

The list of numbers

represents a geometric sequence because, beginning with the first term, $3$ , each term is being multiplied by $5$ to get the next term in the sequence.

The next term in the sequence will be ( $1875 \times 5$ ) or $9375$ .

The number being multiplied each time is called the common ratio (r).

The sequence can be represented by a recursive equation.

In words:

Name the $first term$ . $Next = Previous \times r$

Using function notation:

A geometric sequence can also be represented with an explicit equation in the form $f (x) = a b^{n}$ , where $a$ is the first term, $b$ is the common ratio ( $r$ ) and $n$ is the input value.

The explicit equation for a geometric sequence is an exponential function.

The graph of the terms in a geometric sequence is arranged in a curve.

half-plane

The part of the plane on one side of a straight line of infinite length in the plane.

The points in a half-plane are solutions to an inequality.

histogram

A graphical display of univariate data. The data is grouped into equal ranges and then plotted as bars. The height of each bar shows how many are in each range.

The graph shows the heights of $25$ students in a math class.

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 $x$ 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, $y = - 3$ .

M–R

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.

mean

See measures of central tendency.

mean absolute deviation - M.A.D

M.A.D of a data set is the average distance between each data value and the mean. Mean absolute deviation is a way to describe variation in a data set. It tells us how far, on average, all values are from the middle. There are 3 steps for finding the M.A.D.

Find the mean of all the values.
Find the distance of each value from the mean. (Recall distance is +.)
Find the mean of those distances.

measures of central tendency

A single value that describes the way in which a group of data cluster around a central value. The most common measures of central tendency are the arithmetic mean, the median and the mode.

The mean (average) is found by adding all of the numbers together and dividing by the number of items in the set:

Example: $(10 + 10 + 20 + 40 + 70) / 5 = 30$ .

The median is found by ordering the set from lowest to highest and finding the exact middle. The median is just the middle number: 20.

To calculate the mode, put the numbers in order. Then count how many of each number. The number that appears most is the mode. There may be no mode if no value appears more than any other. There may also be two modes (bimodal), three modes (trimodal), or four or more modes (multimodal).

median

See measures of central tendency.

midpoint

A point on a line segment that divides it into two equal parts.

The formula for finding half the distance between two points (or the midpoint $M$ ) in a coordinate grid is:

$M = (\frac{x_{1} + x_{2}}{2}, \frac{y_{1} + y_{2}}{2}) .$

$M = (\frac{- 3 + 2}{2}, \frac{- 2 + 4}{2}) = (- 0.5, 1)$

See also bisect.

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).

See measures of central tendency.

multi-step equation

An equation for which multiple inverse operations will have to be applied, in the correct order, to solve the equation for its variable.

observed value

The value that is actually observed (what actually happened).

obtuse angle / obtuse triangle

opposite (or negative) reciprocal slope

Slopes of perpendicular lines are opposite reciprocals, so that the product of the slopes is $- 1$ . (See perpendicular lines)

outliers

Values that stand away from the body of the distribution. For a box-and-whisker plot, points are considered outliers if they are more than 1.5 times the interquartile range (length of box) beyond quartiles 1 and 3. A point is also considered an outlier if it is more than two standard deviations from the center of a normal distribution.

parabola

The graph of every equation that can be written in the form $y = A x^{2} + B x + C$ , where $A \geq 1$ is in the shape of a parabola. It looks a bit like a U but it has a very specific shape. Moving from the vertex, it is the exact same shape on the left as it is on the right. (It is symmetric.)

The graph of the parent function or

$y = x^{2}$ follows the pattern:

move right 1 step, move up $1^{2}$ or $1$
move right 2 steps, move up $2^{2}$ or $4$
move right 3 steps, move up $3^{2}$ or $9$

parallel line

perpendicular lines

Two lines or line segments are perpendicular if they have opposite, reciprocal slopes, or if one is vertical and the other is horizontal. Two lines are perpendicular if their intersection forms four right ( $90 °$ ) angles.

plane

A plane is an undefined term because it is an abstract idea rather than concrete like a piece of paper. A plane has two dimensions. It can be identified by determining three noncollinear points. It is labeled according to the letters used to label the points, such as plane $A B C$ .

point

A point is an undefined term because it is an abstract idea rather than concrete like a dot. A point in geometry is a location. It has no size, (i.e., no width, no length, and no depth). A point is labeled with a dot and a capital letter.

point-slope form of a line

You need the slope and a point. Let $slope = m$ and use point $(x_{1}, y_{1})$

The traditional way: $y - y_{1} = m (x - x_{1})$

If we use a property of equality and add $y_{1}$ to both sides of the equation, we get an equation that is more useful:

$y = m (x - x_{1}) + y_{1}$

polynomial function

A function of the form:

$y = a_{0} x^{n} + a_{1} x^{n - 1} + a_{2} x^{n - 2} + \dots a_{n - 3} x^{3} + a_{n - 2} x^{2} + a_{n - 1} x + a_{n}$

where all of the exponents are positive integers and all of the coefficients $a_{0} . . . a_{n}$ are constants.

prime number

A prime number is a positive integer that has exactly two positive integer factors, $1$ and itself. That means $1$ is not a prime number, because it only has one factor, itself. Here is a list of all the prime numbers that are less than $20 : 2, 3, 5, 7, 11, 13, 17, 19$ .

profit

Profit, typically called net profit, is the amount of income that remains after paying all expenses, debts, and operating costs.

properties of equality

The properties of equality describe operations that can be performed on each side of the equal sign ( $=$ ) and still ensure that the expressions remain equivalent.

In the table below, $a$ , $b$ , and $c$ stand for arbitrary numbers in the rational, real, or complex number systems. The properties of equality are true in these number systems.

Reflexive property of equality	$a = a$
Symmetric property of equality	If $a = b$ , then $b = a$
Transitive property of equality	If $a = b$ and $b = c$ , then $a = c$
Addition property of equality	If $a = b$ , then $a + c = b + c$
Subtraction property of equality	If $a = b$ , then $a - c = b - c$
Multiplication property of equality	If $a = b$ , then $a \times c = b \times c$
Division property of equality	If $a = b$ and $c \neq 0$ , then $a \div c = b \div c$
Substitution property of equality	If $a = b$ , then $b$ may be substituted for $a$ in any expression containing $b$

properties of inequality

In the table a, b, and c stand for arbitrary numbers in the rational, or real number systems. The properties of inequality are true in these number systems.

Exactly one of the following is true: $a < b$ , $a = b$ , $a > b$

If $a > b$ and $b > c$ then $a > c$

If $a > b$ , then $b < a$

If $a > b$ , then $a \pm c > b \pm c$

If $a > b$ and $c > 0$ , then $a \cdot c > b \cdot c$

If $a > b$ and $c < 0$ , then $a \cdot c < b \cdot c$

If $a > b$ and $c > 0$ , then $a \div c > b \div c$

If $a > b$ and $c < 0$ , then $a \div c < b \div c$

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

The letters $a$ , $b$ , and $c$ 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	$(a + b) + c = a + (b + c)$
Commutative Property of addition	$a + b = b + a$
Additive identity property of $0$	$a + 0 = 0 + a = a$
Existence of additive inverses	For every $a$ , there exists $- a$ , so that $a + (- a) = (- a) + a = 0$ .
Associative property of multiplication	$(a \cdot b) \cdot c = a \cdot (b \cdot c)$
Commutative Property of multiplication	$a \cdot b = b \cdot a$
Multiplicative identity property of $1$	$a \cdot 1 = 1 \cdot a = a$
Existence of multiplicative inverses	For every $a \neq 0$ , there exists $\frac{1}{a}$ , so that $a \cdot \frac{1}{a} = \frac{1}{a} \cdot a = 1$ .
Distributive property of multiplication over addition	$a (b + c) = a \cdot b + a \cdot c$

quadratic equations

An equation that can be written in the form $y = a x^{2} + b x + c, where a \geq 1$

Standard form: $f (x) = a x^{2} + b x + c, where a \neq 0$

Example: $x^{2} - 2 x - 8 = 0$

Factored form: $(x - 4) (x + 2) = 0$

Vertex form: ${(x - 1)}^{2} - 9 = 0$

Recursive form: $f (1) = - 9; f (n) = f (n - 1) + (2 n - 3)$

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

quadratic function

quantitative variable

Also called numerical data, these data have meaning as a measurement or a count. Can be discrete data representing items that can be listed out or continuous data whose possible values cannot be counted and can only be described using intervals.

quantity

A quantity is an amount, number, or measurement. It answers the question “How much?”

range (statistics)

The difference between the highest and lowest values. It’s one number.

Example: ${3, 22, 17, 1, 9}$ The highest number is $22$ and the lowest is $1$ . The $range$ is $22 - 1 = 21$ .

range of a function

All the resulting $y$ -values obtained after substituting all the possible $x$ -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

A rate that describes how one quantity changes in relation to another quantity. In a linear function the rate of change is the slope. In an exponential function the rate of change is called the change factor or growth factor. Quadratic functions have a linear rate of change (the change is changing in a linear way.)

ratio

A ratio compares the size or amount of two values.

Here is a sentence that compares apples to oranges as shown in the diagram below: “We have five apples for every three oranges.” It describes a ratio of $5$ to $3$ or $5 : 3$ . A ratio can also be written as a fraction, in this case $\frac{5}{3}$ .

Compare oranges to apples. The ratio changes to $3 : 5$ or $\frac{3}{5}$ .

The two previous ratios are called part-to-part ratios. Another way to write a ratio is to compare a part to a whole.

Compare apples to the total amount of fruit. The ratio changes to $5 : 8$ or $\frac{5}{8}$ .

Ratios can be scaled up or down. There are $7$ bags of fruit, each containing $3$ oranges and $5$ apples. The ratio $5$ to $3$ still represents the number of apples compared to the number of oranges. But the ratio $35$ to $21$ also compares the number of apples to the number of oranges.

reciprocal

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 $x$ -coordinate; the second is the $y$ -coordinate.

recursive equation

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

recursive thinking

Noticing the relationship of one output to the next output.

regression line (statistics)

Also called the line of best fit. The line is a model around which the data lie if a strong linear pattern exists. It shows the general direction that a group of points seem to follow. The formula for the regression line is the same as the one used in algebra $y = m x + b$ .

representations

Mathematical representations are tools for thinking about and organizing information in a situation. Representations include tables, graphs, different types of equations, stories or context, diagrams, etc.

residuals, residual plot

The difference between the observed value (the data) and the predicted value (the $y$ -value on the regression line). Positive values for the residual (on the $y$ -axis) mean the prediction was too low, and negative values mean the prediction was too high; $0$ means the guess was exactly correct.

Create a scatter plot and graph the regression line. Draw a line from each point to the regression line, like the segments drawn in blue.

A residual plot highlights:

How far the data is from the predicted value.
Possible outliers
Patterns in the data that suggest a different type of model
If a linear model fits the data.

revenue

Revenue is the total amount of income generated by the sale of goods or services related to the company’s primary operations.

right angle

An angle that measures $90 °$ .

The symbol for a right angle in a geometric figure is a box.

S–X

satisfies an equation

See solution to an equation.

scalene triangle

A triangle that has three unequal sides.

scatter plot

A display of bivariate data (ordered pairs) organized into a graph. A scatter plot has two dimensions, a horizontal dimension ( $x$ -axis) and a vertical dimension (the $y$ -axis). Both axes contain a number line.

secant line

Also simply called a secant, is a line passing through two points of a curve. The slope of the secant is the average rate of change over the interval between the two points of intersection with the curve.

sequence

A list of numbers in some sort of pattern: patterns could be arithmetic, geometric, or other.

set

A set is a collection of things. In mathematics it’s usually a collection of numbers. When writing sets in mathematics, the numbers are listed inside of brackets { }. This is the set of the first five counting numbers. ${1, 2, 3, 4, 5}$

set builder notation

A notation for describing a set by listing its elements or stating the properties that its members must satisfy.

The set ${x | x > 5}$ is read aloud as “the set of all $x$ such that $x$ is greater than $5$ .”

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).

slope

A linear function has a constant slope $(m)$ 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 $(x_{1}, y_{1})$ and $(x_{2}, y_{2})$

$m = \frac{(y_{2} - y_{1})}{(x_{2} - x_{1})}$

$m$ is the symbol for slope.

slope-intercept form of a line

An explicit equation for a line that uses the $slope (m)$ and the $y -intercept (b)$ .

solution set for the system of inequalities

The set of points that satisfy all of the inequalities in a system simultaneously.

Example: The solution for a system is $y > x + 5$ and $y \leq - \frac{3}{4} x$ .

Each inequality in the solutions is graphed. The solution set is the triangle where the blue and green overlap. This is the region where each ordered pair $(x, y)$ within the region makes each inequality true.

See also: Compound inequality in two variables.

If the system represents the constraints in a modeling context, then the feasible region is the set of viable options within the solution set that satisfy all of the constraints simultaneously.

solution to an equation (satisfies an equation)

The value of the variable that makes the equation true.

solve a system by elimination or substitution

See system of equations.

special products of binomials

Some products occur often enough in Algebra that it is advantageous to recognize them by sight. Knowing these products is especially useful when factoring. When you see the products on the right, think of the factors on the left.

spread of a distribution (statistics)

Measures of spread describe how similar or varied the set of observed values are for a particular variable (data item). Measures of spread include the range, quartiles and the interquartile range, variance and standard deviation.

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 $σ$ . (sigma)

standard form of line

$A x + B y = C$

where $A$ , $B$ , and $C$ are integers and $A > 0$ .

subtrahend

See subtraction of polynomials.

symmetric

If a figure can be folded or divided in half so that the two halves match exactly, then such a figure is called a symmetric figure. The fold line is the line of symmetry.

symmetric distribution

Where most of the observations cluster around the central peak. The mean, median, and mode are all equal.

system of equations

A set of two or more equations with the same set of unknowns (or variables), meaning that the solutions for the variables are the same in each of the equations in the set.

Example: The equations for this system are $y = x + 2$ and $y = \frac{- 1}{2} x + 5$ . This can be solved in three ways:

First, observe that both equations are equal to $y$ . This means that the first equation can be substituted for $y$ in the second equation, giving the equation $x + 2 = \frac{- 1}{2} x + 5$ . Solving this shows that $x = 2$ , and by substituting $2$ for $x$ this means that $y = 4$ , so the solution for this system of equations is $(2, 4)$ . This method is called substitution.

Second, these equations can be manipulated so that one variable can be eliminated. In this system, the second equation can be multiplied by $2$ . $2 (y = \frac{- 1}{2} x + 5)$ turns into $2 y = - 1 x + 10$ . This is then added to the first equation:

$2 y = - 1 x + 10$

$\underset{―}{+ y = 1 x + 2}$

$3 y = 0 x + 12$

Since the $x$ and $- x$ subtract to be $0$ , the remaining equation has one variable which can be solved to show that $y = 4$ . Once again, plugging this value into either of the other equations will give $x = 2$ . This method is called elimination.

Finally, both equations can be graphed. The point at which they intersect, as shown below, is the solution to the system of equations.

system of inequalities

A set of two or more inequalities with the same variables. The solution to an inequality includes a range of values. The solution to a system of inequalities is the intersection of all of the solutions. See feasible region.

systems: inconsistent / independent

When a system of equations has no solution, it is called inconsistent. If a system of equations is inconsistent, then when we try to solve it, we will end up with a statement that isn’t true such as $7 = 0$ .The graphs of the equations never intersect.

A system of equations is considered independent if the graphs of the equations create different lines. Independent systems of equations have one solution that can be found graphically or algebraically.

tangent to a curve

A line that touches a curve in exactly one point.

As the two points that form a secant line are brought together (or the interval between the two points is shortened), the secant line tends to a tangent line.

trinomial

A polynomial with three terms.

uniform distribution

A uniform distribution is evenly spread with no obvious mode.

units

A measurement unit is a standard quantity used to express a physical quantity. It identifies the items that are being counted. Units could be inches, feet, or miles. Units could also be oranges, bicycles, or people.

univariate data

Describes a type of data which consists of observations on only a single characteristic or attribute. It doesn’t deal with causes or relationships (unlike regression) and it’s major purpose is to describe. A histogram displays univariate data.

variability

Refers to how spread out a group of data is. Values that are close together have low variability; values that are spread apart have high variability.

variable (algebra)

A symbol for a number we don’t know yet, usually a lowercase letter, often $x$ or $y$ . If a variable is used twice in the same expression, it represents the same value. A number by itself is called a constant. A coefficient is a number used to multiply a variable.

variable (statistics)

A characteristic that’s being counted, measured, or categorized.

vertex

See angle.

vertex of a parabola

Either the maximum or the minimum point of a parabola.

viable, non-viable options

Viable options are values that work in all of the equations in a system. Non-viable options don’t work in all of the equations.

x-intercept

The point(s) where a line or a curve cross the $x$ -axis. The $y$ -value of the point will be $0$ . A non-horizontal line will only cross the $x$ -axis once. A curve could cross the $x$ -axis several times.

Y–Z

y-intercept: The point where a line or a curve crosses the $y$ -axis. The $x$ -value of the point will be $0$ . The $y$ -intercept is often referred to as “ $b$ ” when writing the equation of a line or as the point $(0, b)$ .
A function will have at most one $y$ -intercept.