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

A–F

AA similarity theorem

Two triangles are similar if they have two corresponding angles that are congruent.

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:

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

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

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

adjacent

adjacent angles

Two non-overlapping angles with a common vertex and one common side.

$∠ 1$ and $∠ 2$ are adjacent angles:

alternate exterior angles

A pair of angles formed by a transversal intersecting two lines. The angles lie outside of the two lines and are on opposite sides of the transversal.

See angles made by a transversal.

alternate interior angles

A pair of angles formed by a transversal intersecting two lines. The angles lie between the two lines and are on opposite sides of the transversal.

G–L

geometric mean

A special type of average where $n$ numbers are multiplied together and then the $n th$ root is taken. For two numbers, the geometric mean would be the square root. For three numbers, it would be the cube root.

Example: The geometric mean of $4$ and $9$ is $\sqrt{4 \cdot 9} = 6$ .

The geometric mean of two numbers $a$ and $c$ is the number $b$ such that $\frac{a}{b} = \frac{b}{c}$ .

geometric series

The sum of the terms in a geometric sequence represented by summation notation $\sum_{k = 1}^{n} f (k)$ .

Example: $\sum_{k = 1}^{4} 3 k = (3 \cdot 1) + (3 \cdot 2) + (3 \cdot 3) + (3 \cdot 4) = 30$

horizontal shift

See transformations on a function.

hyperbola

A hyperbola is the set of all points such that the difference of the distances to the foci is constant.

Equation: $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$

hypotenuse

The longest side in a right triangle.

The side opposite the right angle.

imaginary number

See complex number.

incenter

The point of intersection of the angle bisectors in a triangle is the incenter. Each point on the angle bisector is equidistant from the sides of the angle.

The point at which all the three angle bisectors meet is the center of the incircle.

independent event / dependent event

When two events are said to be independent of each other, the probability that one event occurs in no way affects the probability of the other event occurring.

When you flip two coins, each flip is an independent event.

An event is dependent if the occurrence of the first event affects the occurrence of the second so that the probability is changed.

Example: Suppose there are $5$ balls in a box. What is the chance of getting a green ball out of the box on the first try? A green ball is selected and removed in event $1$ . What is the chance of getting a green ball on the second try?

inscribed angle

See angles associated with circles.

inscribed in a circle

intersection of sets

The intersection of two sets $X$ and $Y$ , is the set containing all of the elements of $X$ that also belong to $Y$ . The symbol for intersection is $\cap$ .

For example: If $X = {5, 6, 7, 8, 9, 10}$ and $Y = {5, 10, 15}$ then $X \cap Y = {5, 10}$ .

inverse trigonometric ratio

The inverse of a trigonometric function is used to obtain the measure of an angle when the trigonometric ratio is known.

Example: The inverse of sine is denoted as arcsine, or on a calculator it will appear as $\sin^{- 1}$ .

If $\sin A = \frac{\sqrt{3}}{2}$ and the measure of the angle $A$ is needed, write $\sin^{- 1} (\frac{\sqrt{3}}{2})$ to express this. The answer to the expression is the measure of the angle.

$\sin^{- 1} (\frac{\sqrt{3}}{2}) = 60 °$

All of the inverse trigonometric functions are written the same way.

$\sin^{- 1} (\frac{o}{h}) = A, \cos^{- 1} (\frac{a}{h}) = A, \tan^{- 1} (\frac{o}{a}) = A$

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. $x + (- x) = 0$ . For every $a$ there exists $- a$ so that

$a + (- a) = (- a) + a = 0$

The reciprocal of a nonzero number is the multiplicative inverse of that number. The reciprocal of $x$ is $\frac{1}{x}$ because $x \cdot \frac{1}{x} = 1$ . The product of a real number and its multiplicative inverse is $1$ . Every real number has a unique multiplicative inverse.

joint events

Events that can occur at the same time.

Two-way tables show joints. See two-way tables.

kite

A quadrilateral with two pairs of congruent, adjacent sides.

limit (convergence)

Sometimes in math we can see that an output is getting closer and closer to a value. We can also see that the output won’t exceed this value. We call this a limit.

Example 1: As $n$ gets larger, the value of $n \cdot \sin (\frac{360 °}{2 n}) \cdot \cos (\frac{360 °}{2 n})$ is getting very close to the value of $π$ . We say $π$ is the limit.

Example 2: The more sides in a polygon, the closer the polygon gets to being a circle. The circle is the limit.

More formally: A repeated calculation process that approaches a unique value, called the limit.

line of symmetry

The vertical line that divides the graph into two congruent halves, sometimes called axis of symmetry.

The equation for the line of symmetry in a coordinate plane is always:

$x = the number that the line of symmetry passes through$

linear pair

Two supplementary angles that share a vertex and a side.

A linear pair always make a line.

M–R

major axis, minor axis of an ellipse

The major axis is the longest diameter of an ellipse. It goes from one side of the ellipse, through the center, to the other side, at the widest part of the ellipse. $C D$ is the major axis.

The minor axis is the shortest diameter (at the narrowest part of the ellipse).

median in a triangle

A line segment in a triangle that extends from any vertex to the midpoint of the opposite side.

midline of a triangle

$\overset{―}{D E}$ is the midline of $△ A B C$ .

midline of a triangle theorem

The midline of a triangle or the midsegment theorem states that the segment connecting the midpoints of two sides of a triangle is parallel to the third side and half as long as the third side.

model, mathematical

Modeling with mathematics is the practice of making sense of the world through a mathematical perspective. A mathematical model could be an equation, graph, diagram, formula, sketch, computer program, or other representation that will help you to study different components of a function or to make predictions about behavior.

modulus

The modulus of the complex number $a + b i$ is $| a + b i | = \sqrt{a^{2} + b^{2} .}$ This is the distance between the origin $(0, 0)$ and the point $(a, b)$ 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:

$a + b i$ and $c + d i$ , the distance between them is

$d = \sqrt{{(c - a)}^{2} + {(d - b)}^{2}}$

Find the distance between $1 + 3 i$ and $- 2 - 1 i$ .

$d = \sqrt{{(- 2 - 1)}^{2} + {(- 1 - 3)}^{2}} = \sqrt{9 + 16} = 5$

mutually exclusive

Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint. If two events are disjoint, then the probability of them both occurring at the same time is 0.

mutually exclusive event

Both events can’t happen at the same time. It must be one or the other, but not both.

Example: heads and tails are mutually exclusive when flipping a coin.

n-gon

A polygon with $n$ number of sides.

See polygon.

opposite side in a triangle

A side opposite an angle in a triangle is the side that is not part of the angle.

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 $9$

parabola: conic definition, geometric definition

A parabola is the set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix . The directrix is perpendicular to the axis of symmetry of a parabola and does not touch the parabola.

perpendicular bisector

The line (line segment or ray) that divides a line segment into two equal lengths and makes a right angle with the line segment it divides.

piece-wise defined function

A function which is defined by two or more equations, each valid on its own interval. A piecewise function can be continuous or not.

Each equation in a piece-wise defined function is called a sub-function.

point of concurrency

See concurrent lines.

postulate

A simple and useful statement in geometry that is accepted by the mathematical community as true without proof.

prism: right, oblique

Prism: Also called a polyhedron.

A solid object with two identical ends and flat sides. The ends (bases) are parallel. The shape of the ends gives the prism its name, such as triangular prism or square prism. The sides are parallelograms.

proof: types—flow, two-column, paragraph

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 $b$

proportion: proportionality statement

A proportion is a statement that two ratios are equal.

pyramid

A 3-D shape that has a base, which can be any polygon, and three or more triangular faces that meet at a point called the apex.

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 formula

The quadratic formula allows us to solve any quadratic equation that’s in the form $a x^{2} + b x + c = 0$ . The letters $a$ , $b$ , and $c$ in the formula represent the coefficients of the terms.

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$

quadratic function

quadratic inequality

A function whose degree is $2$ and where the $y$ 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 $\leq$ , and greater than or equal to $\geq$ .

Example: $y \geq a x^{2} + b x + c$

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.

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.

reasoning – deductive/inductive

Two Types of Reasoning
Inductive reasoning: from a number of observations, a general conclusion is drawn.	Deductive reasoning: from a general premise (something we know), specific results are predicted.
Observations	General Premise
Each time I make two lines intersect, the opposite angles are congruent. I have tried this 20 times and it seems to be true. Conclusion: Opposite angles formed by intersecting lines are always congruent.	Given: Angles 1, 2, 3, and 4 are formed by two intersecting lines. Prove: Opposite angles formed by intersecting lines are always congruent.

Two Types of Reasoning

Inductive reasoning:

from a number of observations, a general conclusion is drawn.

Deductive reasoning:

from a general premise (something we know), specific results are predicted.

Observations

General Premise

Each time I make two lines intersect, the opposite angles are congruent. I have tried this 20 times and it seems to be true.

Conclusion:

Opposite angles formed by intersecting lines are always congruent.

Given: Angles 1, 2, 3, and 4 are formed by two intersecting lines.

Prove: Opposite angles formed by intersecting lines are always congruent.

reference angle

The acute angle between the terminal ray of an angle in standard position and the $x$ -axis.

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.

roots: real and imaginary

The solutions of an equation in the form $f (x) = 0$ .

S–X

same-side interior angles

See angles made by a transversal.

SAS triangle similarity

See triangle similarity.

scale factor

The ratio of any two corresponding lengths in two similar geometric figures.

secant line (in a circle), tangent line

Secant line: A line that intersects a circle at exactly two points.

Tangent line: A line that intersects a circle at exactly one point.

sector

The part of a circle enclosed by two radii of a circle and their intercepted arc.

A pie-shaped part of a circle.

segment of a circle

A segment of a circle is a region in a plane that is bounded by an arc of a circle and by the chord connecting the endpoints of the arc.

side-splitter theorem

The side-splitter theorem is related to the midline of a triangle theorem. It extends the rule to say if a line intersects two sides of a triangle and is parallel to the third side of the triangle, it divides those two sides proportionally.

similarity

A 2-D figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations.

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.

SSS triangle similarity

See triangle similarity.

standard form of a quadratic function

$a x^{2} + b x + c, a \neq 0$

straight angle

When the legs of an angle are pointing in exactly opposite directions, the two legs form a single straight line through the vertex of the angle. The measure of a straight angle is always $180 °$ . It looks like a straight line.

sub-function

See piece-wise defined function.

summation notation

supplementary angles

Two angles whose measures add up to exactly $180 °$ .

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.

symmetry

A line that reflects a figure onto itself is called a line of symmetry.

A figure that can be carried onto itself by a rotation is said to have rotational symmetry.

tessellation

A tessellation is a regular pattern made up of flat shapes repeated and joined together without any gaps or overlaps. Many regular polygons tessellate, meaning they can fit together without any gaps.

theorem

A theorem is a statement that can be demonstrated to be true by using definitions, postulates, properties, and previously proven theorems.

The process of showing a theorem to be correct is called a proof.

transformations on a function (non-rigid)

A dilation is a nonrigid transformation $k f (x)$ because the shape changes in size. It will make the function change faster or slower depending on the value of $k$ . If $k > 1$ , it will grow faster and look like it has been stretched. If $0 < k < 1$ , the function will change more slowly and will appear to be fatter. A dilation is also called a vertical stretch.

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

$f (x) + k$

Up when $k > 0$

Down when $k < 0$

Horizontal shift

$f (x + k)$

Left when $k > 0$

Right when $k < 0$

Reflection

$- f (x)$ reflection over the $x$ -axis

$f (- x)$ reflection over the $y$ -axis

A dilation is a nonrigid transformation. It will make the function changes faster or slower depending on the value of $k f (x)$ . If $k > 1$ , it will grow faster and look like it has been stretched. If $0 < k < 1$ , the function will grow more slowly and will appear to be fatter.

transversal

A line that passes through two lines in the same plane at two distinct points. The two lines do not need to be parallel. But when the lines are parallel, several special angle relationships are formed.

tree diagram

A tool in probability and statistics used to calculate the number of possible outcomes of an event, as well as list those possible outcomes in an organized manner.

triangle similarity

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion. Similar triangles are the same shape, but not necessarily the same size.

There are three similarity patterns that provide sufficient information to prove two triangles are similar:

AA Similarity

SAS Similarity

SSS Similarity

trigonometric ratios in right triangles: sine A, cosine A, tangent A

An operation that relates the measure of an angle with a ratio of the lengths of the sides in a right triangle. There are three trigonometric ratios, plus their reciprocals. See Reciprocal trigonometric functions for definitions.

$sine A$ abbreviated $\sin A$

$cosine A$ abbreviated $\cos A$

$tangent A$ abbreviated $\tan A$

A trigonometric ratio always includes a reference angle.

In right triangle $A B C$ , the trigonometric ratios are defined as:

$\sin A = \frac{the length of the opposite side}{the length of the hypotenuse} = \frac{a}{c}$

$\cos A = \frac{the length of the adjacent side}{the length of the hypotenuse} = \frac{b}{c}$

$\tan A = \frac{the length of the opposite side}{the length of the adjacent side} = \frac{a}{b}$

Note that each trigonometric function above references the angle $A$ . If angle $B$ was referenced as the angle, the opposite and adjacent sides would be in reference to angle $B$ , and they would switch sides.

trinomial

A polynomial with three terms.

two-column proof

See proof:types—flow, two-column, paragraph.

two-way table

A table listing two categorical variables whose values have been paired such that the possible values of one variable make up the rows and the possible values for the other variable make up the columns. The green cells on this table are where the joint frequency numbers are located. They are called joint frequency because you are joining one variable from the row and one variable from the column. The marginal frequency numbers are the numbers on the edges of a table. On this table, the marginal frequency numbers are in the purple cells.

union

The union of two sets is a set containing all elements that are in set $A$ or in set $B$ (or possibly both). The symbol for union is $\cup$ .

For example, ${5, 8, 11} \cup {6, 7, 9, 10} = {5, 6, 7, 8, 9, 10, 11}$ .

vertex

See angle.

vertex form

See quadratic function.

vertex of a parabola

Either the maximum or the minimum point of a parabola.

vertical angles

The angles opposite each other when two lines cross. They are always congruent.

vertical shift

See transformations on a function (rigid).

vertical stretch

See transformations on a function (non-rigid).

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

zero property of multiplication (also called the zero product property)
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 $x$ -intercepts of the graph of a function.