Weeks 1-5 (1-19) Rules and Definitions Shormann Algebra 2

relation

Like a function, except the input (x), can map to more than one output (y).

Euclid's 5 postulates

P1: Two, and only two points determine one unique straight line. P2: A straight line extends an indefinite length in either direction. P3: A circle may be drawn with any given center and any given radius. P4: All right angles are equal to one another. P5: (Parallel postulate) Given a line n and a point P not on that line, there exists in the plane of P and n and through P only one line m, which does not intersect line n.

Rate = distance ÷ time

R = D/T

tangent

The trigonometric function that is equal to the ratio of the side opposite a given angle (in a right triangle) to the adjacent side

sine

The trigonometric function that is equal to the ratio of the side opposite a given angle (in a right triangle) to the hypotenuse.

The measure of an arc equals the measure of its central angle

There are 360° of arc in a circle.

Triangle Congruency Theorems

Third angle theorem: If two angles in one triangle are congruent to two angles in another triangle, the third angles are also congruent.

coterminal angles

Two angles are coterminal if both have their terminal rays in the same location. For example, drawn on an x-y axis, and beginning on the positive x-axis, a 30° angle would be coterminal with a 30 + 360°, 30 + 2(360°), 30 + 3(360°), ..... 30 + n(360°) angle.

system of equations

Two or more equations (or functions, or relations), with two or more unknowns, that are solved together. A system of equations always has the same number of equations as unknowns.

Angles

When a transversal intersects two or more parallel lines, all acute angles are equal, and all obtuse angles are equal.

indeterminate

When both the numerator and denominator equal zero.

analytical geometry

Where algebra and geometry meet on the coordinate plane.

postulate

a construction (drawing) of something, normally common to a particular science, that may not be obvious. Assumed to be true without proof.

axiom

a self-evident statement about something obvious, normally common to all sciences.

polar form

a way to represent a vector by its magnitude (M) and direction (θ). Often written using the notation M∠θ

identity

an equation, usually with only one variable, that is true for any input value.

Circle rules for use in 16D

circumference = 2πr area = πr2

Remembering that mathematics is considered a symbolic language (the language of science), here is a list of operations with pairs of functions, together with notation used to describe them in a more simplified form.

f(x) + g(x) = (f + g)(x) f(x) − g(x) = (f − g)(x) f(x)i g(x) = (fg)(x) f(x) ÷ g(x) = (f / g)(x) We are using f(x) and g(x) because those are just common variables used for functions. The f and g have no special meaning.

Perpendicular lines

have slopes that are negative reciprocals of each other

Parallel lines

have the same slope

point

that which has no part. Its location is represented by a dot. · ← a point

quadrant

the four parts of a coordinate plane

The standard form for the equation of a line

y = mx + b

Volume of any right solid

= Area of base ∙ height

unit circle

A circle with radius = 1 (unity). In trig applications, the unit circle is connected to a right triangle with a hypotenuse = radius =1, or a vector of magnitude = 1 and direction = θ.

data

A collection of facts and information recorded during an experiment, from which conclusions and decisions can be made

undefined

A function is said to be undefined when a value is not part of it's domain. For example, when x equals 0 in f(x) = 1/x. We would say the function is undefined at x = 0 because it's output equals 1/0 = ± ∞, which we cannot define.

inscribed angle

Angle A° in the Inscribed Angle Theorem's diagram. An angle whose vertex touches a circle, forming an intercepted arc on the opposite side. Also defined as two chords having a common endpoint.

Triangle Congruency Theorems

Angle-angle-angle-side (AAAS): If the angles in one triangle equal the angles in a second triangle, the triangles are similar. If it is also known that at least one pair of sides opposite the same angle measure are congruent, then the two triangles are also congruent. Also known as ASA (because if two angles are congruent, then so are the third angles), it is based off of Euclid's Proposition 26.

right solid

Any geometric solid whose sides are perpendicular to its base.

minor arc

Arc x° in the Inscribed Angle Theorem's diagram.

major arc

Arc y° in the Inscribed Angle Theorem's diagram.

Distance equals rate × time

D = RT

Angles

- All right angles equal 90° - All straight angles equal 180° (two right angles) - When two line segments intersect, the vertical (opposite) angles are equal.

Polygons

- For any polygon, the number of sides equals the number of vertices. - For any convex polygon of N sides, the sum of the measures of the interior angles equals (N - 2)180°. - For any convex polygon of N sides, the sum of the measures of the exterior angles equals 360°.

Triangles

- In any triangle, the sum of the measures of the three angles equals 180°. - In any triangle, the angles opposite sides of equal lengths have equal measures, and vice-versa.

Equivalent measures: Area

1 hectare = 10,000 meters squared 1 Acre = 43,560 feet squared

power function

A function of the form f(x) = axⁿ , where n is a real number. In the graphical examples above, there are actually 5 types of power functions represented;linear (x¹), quadratic (x²), cubic (x³), square root (x½), and reciprocal (x⁻¹).

tangent

A line in the same plane as the circle and touching the circle only once (see Lesson 21 for a similar definition).

chord

A line segment that connects any two points on a circle. Example: a diameter is a chord, but a radius is not.

secant

A line that intersects a circle at two points. Different than a chord because it is a line, not a line segment (To save space, we will usually draw secants as line segments.)

line segment

A line with a start point and end point.

ordered pair

A pair of numbers, written in a specific order. On a coordinate plane, an ordered pair is used to identify the location of a point, and has the form (x,y).

vector

A quantity that has both magnitude (size) and direction (angle measure). Represented on paper as a straight line with an arrow tip.

slope

A ratio expressing the change in the dependent variable (y) with respect to the independent variable (x). Also referred to as "rise over run", "change in y over change in x", and Δy − Δx , where ∆ = "change in."

maxima and minima(known collectively as extrema or optimums)

A real-valued function f(x) defined on a domain D has a global (or absolute) maximum point at x if f(x*) ≥ f(x) for all x in D. Similarly, the function has a global (or absolute) minimum point at x* if f(x*) ≤ f(x) for all x in D. Local extrema that are clearly not the global extrema may also occur in D (see example below).

function

A relationship where the output (y) depends on the input (x). Each input, or domain value, maps to one, and only one value in the output, or range.

continuous function

A very basic definition of a continuous function, f(x), is a function where a small change in x results in a small change in f(x), with no sudden jumps or gaps. If there are sudden jumps or gaps, then the function is said to be discontinuous, or to have a discontinuity. Also, if you are drawing a graph of the function, it is discontinuous if you have to pick up your pencil in order to draw two or more separate portions.

line

A widthless length. Its location is represented on paper by using a pencil and straight edge. ↔

Euclid's 5 axioms, or common notions:

A1: Things that are equal to the same thing are also equal to one another. If a = c and b = c, then a = b A2: If equals be added to equals, the wholes are equal. If a = b and c = d, then a + c = b + d A3: If equals be subtracted from equals, the remainders are equal. If a = b and c = d, then a - c = b - d A4: Things which coincide with one another are equal to one another. In other words, a = a (reflexive axiom) A5: The whole is greater than the part.

Q.E.D.

Acronym for the Latin phrase quod erat demonstrandum, "that which had to be demonstrated." Used to signal the end of a mathematical proof.

cartesian coordinate system

Also called a coordinate plane, it is a plane containing a horizontal, "x"-axis and vertical, "y"-axis. It is used to graph ordered pairs, functions, experimental data, etc.

trigonometry

From the Greek trigonon (triangle) + metron (measure). The branch of mathematics related to the study of triangles and their side/angle relationships, with special emphasis on right triangles.

Graphing inequalities

Graphs of f(x) > and f(x) < are represented by dashed lines. Graphs of f(x) ≥ and f(x) ≤ are represented by solid lines.

Triangle Congruency Theorems

Hypotenuse-Leg (HL): If the lengths of the hypotenuse and a leg in one right triangle equal the lengths of the hypotenuse and a leg in a second right triangle, the right triangles are congruent.

Triangle Congruency Theorems

Side-angle-side (SAS): If two sides and the included angle in one triangle have the same measures as two sides and the included angle in a second triangle, the triangles are congruent. It is based off of Euclid's proposition 4.

Triangle Congruency Theorems

Side-side-side (SSS): If the lengths of the sides in one triangle are equal to the lengths of the sides in a second triangle, the triangles are congruent. It is based off Euclid's Proposition 8.

Surface Area

Sphere = 4πr² Lateral Surface Area (LSA) of a Cone = πrs LSA of any right solid = Perimeter of base ∙ height

standard of measure

The fundamental reference for a system of weights and measures, against which all other measuring devices are compared. For example, the international prototype of the kilogram, or IPK, is a cylinder made of an alloy of 90% platinum/10% iridium. The IPK was created in 1889, and is stored inside a triple vacuum-sealed container. Copies of the IPK, called "national prototypes" are distributed to interested countries, and calibrated with the IPK about every 40 years.

x-intercept

The location where a function crosses the x-axis.

y-intercept

The location where a function crosses the y-axis.

negative reciprocal

The negative reciprocal of x is 1/x

periodicity

The quality of being periodic; a pattern that repeats itself.

domain

The set of all real number inputs to a function that map to real outputs (range). See also definition of a function in Lesson 12.

range

The set of real number outputs of a function.

Angles

The sum of complementary angles equals 90°. x + y = 90°

Angles

The sum of supplementary angles equals 180°. x + y = 180°

cosine

The trigonometric function that is equal to the ratio of the side adjacent a given angle (in a right triangle) to the hypotenuse.

Intersecting Secants Theorem: When two secant lines intersect each other outside a circle, the products of their segments are equal.

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Secant-Tangent Theorem: If a secant segment and tangent segment are drawn to a circle from the same external point, the product of the length of the secant segment and its external part equals the square of the length of the tangent segment (Euclid's Book 3, Proposition 36).

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Tangent Segments Theorem: Two intersecting tangent segments from points outside a circle have equal lengths

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Rules for converting to and from scientific notation: When converting to scientific notation

moving the decimal place to the left adds positive values to the base 10 exponent. Moving the decimal to the right adds negative values to the base 10 exponent.