Numbers and Divisibility

Rational Numbers

Real numbers/fractions that can repeat or terminate.

Examples: 33, 1/3

Irrational Numbers

Real numbers/fractions that do not repeat or terminate.

Example: π

Integers

Positive or negative whole numbers. 0 is also considered an integer.

Example: 4, -2

Non-Integers

Positive or negative numbers that are in fraction form.

Ex: 25/7

Imaginary Numbers

Numbers that are not real, have an i in them.

Ex:

Divisible by 2

Even #’s

End in 0,2,4,6 or 8

Divisible by 5

Ends in a Zero or Five

Divisible by 10

Ends in Zero

Divisible by 3

Sum digits together

Sum must be divisible by 3

Divisible by 9

Add digits together

Sum of the digits must be divisible by 9

Divisible by 4

If the last two digits are divisible by 4 than the whole number is

Divisible by 6

If its divisible by 2 and 3

Consecutive

• One right after another, the next possible one.

Distinct

• =Different

Factors

• Any group of numbers or variables that when multiplied give the original number/variable

Multiple

• The result of multiplying a number by an integer.

• EX: Multiples of 4:

…,-8,-4,0,4,8,12…

• Union• Combining sets without

writing the repeats

• Intersection• The overlap of sets

Percent Increase or Decrease

100%current original

original

− ×

Exponent and Root Rules!

How to multiply two powers with same base?

a3 * a5 = a3 + 5 = a8

How to divide two powers with the same base?

a5 / a3 = a5-3 = a2

Multiplying exponents

(a2)3= a2*3= a6

Zero as an exponent

a0=1

ANYTHING TO THE ZERO POWER EQUALS 1

Exponent of 1

X1=X

Anything to the exponent of 1, is THAT number

Negative Exponents

a-1= 1/a

Simplifying Radicals with multiplication

Can be written as ba

Simplifying Radicals with division

baba /

Alternate form of square root

= a1/2a

Alternate form of cube root

= a 1/33 a

3 2a = a2/3 = ( )23 a

Graphing/ Writing Equations of Lines

Coordinate Plane

Origin

Y-axis

X-axis

Quadrant IQuadrant 2

Quadrant 3 Quadrant 4

Slope Formula

2 1

2 1

y y risem

x x run

−= =−

Distance Formula

2 22 1 2 1( ) ( )d y y x x= − + −

Midpoint Formula

1 2 1 2,2 2

x x y y+ + = ÷

Vertical Lines

•Think vertebra to help with visual•Undefined Slope! (cannot walk up walls)•Form x=#

Horizontal Lines

•Think horizon to help with visual•Slope = Zero (walking across left to right there is no incline or decline)•Form y=#

Slope-Intercept Form

y mx b= +

Parallel Lines

•Do not intersect•Have the same slopes•Symbol: ||

Perpendicular Lines

•Intersect at a right angle/90⁰•Have slopes that are opposite, reciprocals of each other (flip it and switch it)•Symbol: ⊥

X-intercepts

•Also known as roots and zeros•Where the graph crosses the x-axis•Plug 0 in for y and solve for x•Answer: (#,0) as an ordered pair

y-intercepts

•Where the graph crosses the y-axis•Plug 0 in for x and solve for y•Answer: (0,#) as an ordered pair

Directly Proportional

As x increases, y increasesOR

As x decreases, y decreases

y kx=

Inversely Proportional

As x increases, y decreasesOR

As x decreases, y increases

ky

x=

Function Notationand Variables

Function

• Equation where every input has exactly one output– For each x-value there is one y-value

• F(x)=y– F(x)=mx + b

• Plug in x to find F(x) or y

F(x)=2x+4F(-3)

F(-3)=2(-3)+4

F(-3)=(-6)+4

F(-3)=-2

F(x)=4x+5F(x)=25

25=4x+5

25-5=4x

20=4x

4

X=5

F(x) + G(x)F of x added to G of x

• Add the two functions together

F(x) – G(x)F of x subtracted from G of

x

• Subtract the two functions

F(G(x))F of G of x

• Plug the function of G(x) into the x-variables in the function F(x)

F(x) G(x)●F of x multiplied by G of x

• Multiply the two functions together

F(x) / G(x)F of x divided by G of x

• Divide the two notations

Graph Shiftsf(x)

f(x) + 3

• The f(x) graph moves up 3 places

f(x) - 5

• The f(x) graph moves down 5 places

f(x)

-f(x)

• The f(x) graph is reflected over x-axis

f(-x)

• The graph of f(x) is reflected over the y-axis

f(x + 2)

• The f(x) graph moves LEFT 2

f(x – 4)

• The f(x) graph moves RIGHT 4

Geometry

Sum of Interior Angles of a Triangle?

A

B

C

0180m A m B m C∠ + ∠ + ∠ =

Perimeter of Triangle

a + b + c = perimeter

a b

c

Exterior Angle Theorem

A

B

C D

m A m B m D∠ + ∠ = ∠

Pythagorean Theorem

a

b

C=hypotenuse

2 2 2a b c+ =

Area of a Triangle

Area Formula: ½ x base x height

30⁰-60⁰-90⁰ Right Triangles

60⁰

30⁰

3n

n2n

45⁰-45⁰-90⁰ Right Triangles

n2n

n

45⁰

45⁰

Congruent Triangles

Scalene Triangle

Triangle with no equal sides.

Isosceles Triangle

Triangle with two equal sides. The corresponding angles

are congruent as well.

Equilateral & Equiangular Triangle

(If equilateral equiangular and vice

versa)

Triangle that has three equal sides and three equal angles

that are 60⁰.

Right Triangle

Leg

Hypotenuse

Leg

Obtuse Triangle

Triangle that has one obtuse angle.

Acute Triangle

Triangle that has three acute angles.

Quadrilateral

Four sided Figure

Area of a Quadrilateral

A=base X height

Parallelogram

• Quadrilateral with the following properties:1. Opposite sides are parallel2. Opposite sides are congruent3. Diagonals bisect each other4. Opposite angles are congruent

Rectangle

• Parallelogram that has all of those properties plus the following:1. All angles are 90⁰2. Diagonals are congruent

Rhombus

• Parallelogram that has all of those properties plus the following:1. All sides are congruent2. Diagonals are perpendicular3. Diagonals bisect corner angles

Square

• Parallelogram that has all of those properties plus combines the properties of a rectangle and a rhombus

Sum of Interior Angles of a Polygon

0( 2)180n −

Sum of Exterior Angles of a Polygon

0360

C i cR L e S

Diameter of a circle

d=2rDiameter Radius

Circumference of a circle

C= 2 rCircumference

Radius

Area of a circle

A= r2 Area Radius

Central Angle

Central Angle

O

Arc of a Circle

Arc

O

Sector

• A sector is a region that is formed between two radii and the arc joining their end points

To find the area of a sector…..

360r2

Area of a Circle

Length of Arc

3602 r

Circumference of a Circle

Sum of all angles in a circle

360o

Tangent to a Circle

• Tangent line is perpendicular to the radius at the point of tangency

Probability

Number of favorable outcomeTotal number of outcomes

Statistics Terms

Average=Mean

the sum of a set of values

the total number of values in the set

Median

Middle number in a set of numbers arranged in numerical order

Mean

average of the middle two numbers

Mode

Values that appear the most often in a set of numbers.

Acute Angles

• Angle whose measure is between 0 and 90 degrees.

Obtuse Angles

• Angle whose measure is between 90 and 180 degrees.

Complementary Angles

• Two angles that sum to 90 degrees.

Right Angle

An angle that is 90 degrees

Supplementary Angles

• Two angles that sum to 180 degrees.

Straight Angle

• An angle that’s measure is 180 degrees

Vertical Angles

• Angles that are opposite of each other when two lines cross • Vertical angles are congruent, so angles a and b are congruent in the image.

Transversal

• A line that crosses two lines (they do not have to be parallel) creating special types of angles

Corresponding Angles

• Angles in matching corners are corresponding. • In this image, a and e, b and f, d and h, d and g are

corresponding.• If the transversal crosses two parallel lines, corresponding angles

are then congruent.

Alternate Interior Angles

• The pairs of angles that are on opposite sides of the transversal but inside the other two lines are alternating interior angles

• In this image, c and f, and d and e are alternating interior.• If the transversal crosses two parallel lines, AI angles are then congruent.

Alternate Exterior Angles

• The pairs of angles that are on opposite sides of the transversal but outside the other two lines are alternate exterior angles

• In this image, a and h, and b and g are alternating interior.• If the transversal crosses two parallel lines, AE angles are then congruent.

Same Side Interior Angles

• Angles that are on the same side of the transversal and on the interior of the other two lines are same side interior.

• In this image, 3 and 6, and 4 and 5 are SSI angles.• If the transversal crosses two parallel lines, SSI angles are

supplementary.

Same Side Exterior Angles

• Angles that are on the same side of the transversal and on the exterior of the other two lines are same side exterior.

• In this image, 2 and 7, and 1 and 8 are SSE angles.• If the transversal crosses two parallel lines, SSE angles are

supplementary.