Numbers and Divisibility
Rational Numbers
Real numbers/fractions that can repeator terminate. Examples: 33, 1/3
Irrational Numbers
Real numbers/fractions that do not repeator terminate.                 Example: π
Integers
Positive or negative whole numbers. 0is also considered an integer.              Example: 4, -2
Non-Integers
Positive or negative numbers that are infraction form.                   Ex: 25/7
Imaginary Numbers
Numbers that are not real, have an i inthem.                  Ex:
Divisible by 2
Even #’sEnd 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 togetherSum must be divisible by 3
Divisible by 9
Add digits togetherSum of the digits must bedivisible by 9
Divisible by 4
If the last two digits aredivisible by 4 than the wholenumber 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
current − original                   ×100%    original
Exponent and Root Rules!
How to multiply two powers with         same base?
a *a =a3   5     3+5                =a   8
How to divide two powers with the           same base?
a /a =a =a5   3   5-3   2
Multiplying exponents
(a2)3= a2*3= a6
Zero as an exponent
a0=1ANYTHING TO THE ZERO POWER         EQUALS 1
Exponent of 1
X =X               1Anything to the exponent of 1, is         THAT number
Negative Exponents
a-1= 1/a
Simplifying Radicals with multiplication
Can be written as   a b
Simplifying Radicals with division
aa/b       b
Alternate form of square root
a = a1/2
Alternate form of cube root
3            a = a 1/33   2    a = a     2/3                    =   ( a)                        3                         ...
Graphing/ Writing Equations of           Lines
Coordinate Plane
Y-axisQuadrant 2     Quadrant I                                    X-axis                       Quadrant 4Quadrant 3      ...
Slope Formula
y2 − y1 risem=        =   x2 − x1 run
Distance Formula
d = ( y2 − y1 ) + ( x2 − x1 )                2               2
Midpoint Formula
 x1 + x2 y1 + y2 =        ,        ÷  2          2 
Vertical Lines
•Think vertebra to help with visual•Undefined Slope! (cannot walk upwalls)•Form x=#
Horizontal Lines
•Think horizon to help with visual•Slope = Zero (walking across leftto right there is no incline ordecline)•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 andswitch 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 orde...
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
y = kxAs x increases, y increases            ORAs x decreases, y decreases
Inversely Proportional
k        y=           xAs x increases, y decreases            ORAs x decreases, y increases
Function Notation  and 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     •...
F(x)=2x+4   F(-3)
F(-3)=2(-3)+4 F(-3)=(-6)+4    F(-3)=-2
F(x)=4x+5 F(x)=25
25=4x+525-5=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?                B         A               C
m∠A + m∠B + m∠C = 180   0
Perimeter of Triangle
a + b + c = perimeter     a         b          c
Exterior Angle   Theorem
m∠A + m∠B = m∠D     BA        C   D
Pythagorean Theorem
a +b = c            2          2   2        C=hypotenusea    b
Area of a Triangle
Area Formula: ½ x base x height
30⁰-60⁰-90⁰ Right Triangles
60⁰                2nn                 30⁰          n 3
45⁰-45⁰-90⁰ Right Triangles
45⁰              n 2n                45⁰          n
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 equalsides and three equal angles        that are 60⁰.
Right Triangle
HypotenuseLeg      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 parallel  2. Opposite sides are congruent  3. Diago...
Rectangle
• Parallelogram that has all of those  properties plus the following:  1. All angles are 90⁰  2. Diagonals are congruent
Rhombus
• Parallelogram that has all of those  properties plus the following:  1. All sides are congruent  2. Diagonals are perpen...
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
(n − 2)180   0
Sum of Exterior Angles of a Polygon
0360
C   i        R c Le S
Diameter of a circle
d=2rDiameter          Radius
Circumference of a circle
C= 2 r                RadiusCircumference
Area of a circle
A= r   2Area          Radius
Central Angle
Central AngleO
Arc of a Circle
ArcO
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…..
r2360      Area of a       Circle
Length of Arc
2        r360          Circumference             of a Circle
Sum of all angles in a circle
360   o
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 valuesthe 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 congru...
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...
Alternate Interior Angles
• The pairs of angles that are on opposite sides of the transversal but inside  the other two lines are alternating interi...
Alternate Exterior Angles
• The pairs of angles that are on opposite sides of the transversal but outside  the other two lines are alternate exterio...
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.•...
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.•...
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  1. 1. Numbers and Divisibility
  2. 2. Rational Numbers
  3. 3. Real numbers/fractions that can repeator terminate. Examples: 33, 1/3
  4. 4. Irrational Numbers
  5. 5. Real numbers/fractions that do not repeator terminate. Example: π
  6. 6. Integers
  7. 7. Positive or negative whole numbers. 0is also considered an integer. Example: 4, -2
  8. 8. Non-Integers
  9. 9. Positive or negative numbers that are infraction form. Ex: 25/7
  10. 10. Imaginary Numbers
  11. 11. Numbers that are not real, have an i inthem. Ex:
  12. 12. Divisible by 2
  13. 13. Even #’sEnd in 0,2,4,6 or 8
  14. 14. Divisible by 5
  15. 15. Ends in a Zero or Five
  16. 16. Divisible by 10
  17. 17. Ends in Zero
  18. 18. Divisible by 3
  19. 19. Sum digits togetherSum must be divisible by 3
  20. 20. Divisible by 9
  21. 21. Add digits togetherSum of the digits must bedivisible by 9
  22. 22. Divisible by 4
  23. 23. If the last two digits aredivisible by 4 than the wholenumber is
  24. 24. Divisible by 6
  25. 25. If its divisible by 2 and 3
  26. 26. Consecutive
  27. 27. • One right after another, the next possible one.
  28. 28. Distinct
  29. 29. • =Different
  30. 30. Factors
  31. 31. • Any group of numbers or variables that when multiplied give the original number/variable
  32. 32. Multiple
  33. 33. • The result of multiplying a number by an integer.• EX: Multiples of 4:…,-8,-4,0,4,8,12…
  34. 34. • Union• Combining sets without writing the repeats
  35. 35. • Intersection• The overlap of sets
  36. 36. Percent Increase or Decrease
  37. 37. current − original ×100% original
  38. 38. Exponent and Root Rules!
  39. 39. How to multiply two powers with same base?
  40. 40. a *a =a3 5 3+5 =a 8
  41. 41. How to divide two powers with the same base?
  42. 42. a /a =a =a5 3 5-3 2
  43. 43. Multiplying exponents
  44. 44. (a2)3= a2*3= a6
  45. 45. Zero as an exponent
  46. 46. a0=1ANYTHING TO THE ZERO POWER EQUALS 1
  47. 47. Exponent of 1
  48. 48. X =X 1Anything to the exponent of 1, is THAT number
  49. 49. Negative Exponents
  50. 50. a-1= 1/a
  51. 51. Simplifying Radicals with multiplication
  52. 52. Can be written as a b
  53. 53. Simplifying Radicals with division
  54. 54. aa/b b
  55. 55. Alternate form of square root
  56. 56. a = a1/2
  57. 57. Alternate form of cube root
  58. 58. 3 a = a 1/33 2 a = a 2/3 = ( a) 3 2
  59. 59. Graphing/ Writing Equations of Lines
  60. 60. Coordinate Plane
  61. 61. Y-axisQuadrant 2 Quadrant I X-axis Quadrant 4Quadrant 3 Origin
  62. 62. Slope Formula
  63. 63. y2 − y1 risem= = x2 − x1 run
  64. 64. Distance Formula
  65. 65. d = ( y2 − y1 ) + ( x2 − x1 ) 2 2
  66. 66. Midpoint Formula
  67. 67.  x1 + x2 y1 + y2 = , ÷  2 2 
  68. 68. Vertical Lines
  69. 69. •Think vertebra to help with visual•Undefined Slope! (cannot walk upwalls)•Form x=#
  70. 70. Horizontal Lines
  71. 71. •Think horizon to help with visual•Slope = Zero (walking across leftto right there is no incline ordecline)•Form y=#
  72. 72. Slope-Intercept Form
  73. 73. y = mx + b
  74. 74. Parallel Lines
  75. 75. •Do not intersect•Have the same slopes•Symbol: ||
  76. 76. Perpendicular Lines
  77. 77. •Intersect at a right angle/90⁰•Have slopes that are opposite,reciprocals of each other (flip it andswitch it)•Symbol: ⊥
  78. 78. X-intercepts
  79. 79. •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
  80. 80. y-intercepts
  81. 81. •Where the graph crosses the y-axis•Plug 0 in for x and solve for y•Answer: (0,#) as an ordered pair
  82. 82. Directly Proportional
  83. 83. y = kxAs x increases, y increases ORAs x decreases, y decreases
  84. 84. Inversely Proportional
  85. 85. k y= xAs x increases, y decreases ORAs x decreases, y increases
  86. 86. Function Notation and Variables
  87. 87. Function
  88. 88. • 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
  89. 89. F(x)=2x+4 F(-3)
  90. 90. F(-3)=2(-3)+4 F(-3)=(-6)+4 F(-3)=-2
  91. 91. F(x)=4x+5 F(x)=25
  92. 92. 25=4x+525-5=4x 20=4x 4 X=5
  93. 93. F(x) + G(x)F of x added to G of x
  94. 94. • Add the two functions together
  95. 95. F(x) – G(x)F of x subtracted from G of x
  96. 96. • Subtract the two functions
  97. 97. F(G(x))F of G of x
  98. 98. • Plug the function of G(x) into the x-variables in the function F(x)
  99. 99. F(x) ● G(x)F of x multiplied by G of x
  100. 100. • Multiply the two functions together
  101. 101. F(x) / G(x)F of x divided by G of x
  102. 102. • Divide the two notations
  103. 103. Graph Shiftsf(x)
  104. 104. f(x) + 3
  105. 105. • The f(x) graph moves up 3 places
  106. 106. f(x) - 5
  107. 107. • The f(x) graph moves down 5 places f(x)
  108. 108. -f(x)
  109. 109. • The f(x) graph is reflected over x-axis
  110. 110. f(-x)
  111. 111. • The graph of f(x) is reflected over the y-axis
  112. 112. f(x + 2)
  113. 113. • The f(x) graph moves LEFT 2
  114. 114. f(x – 4)
  115. 115. • The f(x) graph moves RIGHT 4
  116. 116. Geometry
  117. 117. Sum of Interior Angles of a Triangle? B A C
  118. 118. m∠A + m∠B + m∠C = 180 0
  119. 119. Perimeter of Triangle
  120. 120. a + b + c = perimeter a b c
  121. 121. Exterior Angle Theorem
  122. 122. m∠A + m∠B = m∠D BA C D
  123. 123. Pythagorean Theorem
  124. 124. a +b = c 2 2 2 C=hypotenusea b
  125. 125. Area of a Triangle
  126. 126. Area Formula: ½ x base x height
  127. 127. 30⁰-60⁰-90⁰ Right Triangles
  128. 128. 60⁰ 2nn 30⁰ n 3
  129. 129. 45⁰-45⁰-90⁰ Right Triangles
  130. 130. 45⁰ n 2n 45⁰ n
  131. 131. Congruent Triangles
  132. 132. Scalene Triangle
  133. 133. Triangle with no equal sides.
  134. 134. Isosceles Triangle
  135. 135. Triangle with two equal sides. The corresponding angles are congruent as well.
  136. 136. Equilateral & Equiangular Triangle (If equilateral  equiangular and vice versa)
  137. 137. Triangle that has three equalsides and three equal angles that are 60⁰.
  138. 138. Right Triangle
  139. 139. HypotenuseLeg Leg
  140. 140. Obtuse Triangle
  141. 141. Triangle that has one obtuse angle.
  142. 142. Acute Triangle
  143. 143. Triangle that has three acute angles.
  144. 144. Quadrilateral
  145. 145. Four sided Figure
  146. 146. Area of a Quadrilateral
  147. 147. A=base X height
  148. 148. Parallelogram
  149. 149. • Quadrilateral with the following properties: 1. Opposite sides are parallel 2. Opposite sides are congruent 3. Diagonals bisect each other 4. Opposite angles are congruent
  150. 150. Rectangle
  151. 151. • Parallelogram that has all of those properties plus the following: 1. All angles are 90⁰ 2. Diagonals are congruent
  152. 152. Rhombus
  153. 153. • Parallelogram that has all of those properties plus the following: 1. All sides are congruent 2. Diagonals are perpendicular 3. Diagonals bisect corner angles
  154. 154. Square
  155. 155. • Parallelogram that has all of those properties plus combines the properties of a rectangle and a rhombus
  156. 156. Sum of Interior Angles of a Polygon
  157. 157. (n − 2)180 0
  158. 158. Sum of Exterior Angles of a Polygon
  159. 159. 0360
  160. 160. C i R c Le S
  161. 161. Diameter of a circle
  162. 162. d=2rDiameter Radius
  163. 163. Circumference of a circle
  164. 164. C= 2 r RadiusCircumference
  165. 165. Area of a circle
  166. 166. A= r 2Area Radius
  167. 167. Central Angle
  168. 168. Central AngleO
  169. 169. Arc of a Circle
  170. 170. ArcO
  171. 171. Sector
  172. 172. • A sector is a region that is formed between two radii and the arc joining their end points
  173. 173. To find the area of a sector…..
  174. 174. r2360 Area of a Circle
  175. 175. Length of Arc
  176. 176. 2 r360 Circumference of a Circle
  177. 177. Sum of all angles in a circle
  178. 178. 360 o
  179. 179. Tangent to a Circle
  180. 180. • Tangent line is perpendicular to the radius at the point of tangency
  181. 181. Probability
  182. 182. Number of favorable outcomeTotal number of outcomes
  183. 183. Statistics Terms
  184. 184. Average=Mean
  185. 185. the sum of a set of valuesthe total number of values in the set
  186. 186. Median
  187. 187. Middle number in a set of numbers arranged in numerical order
  188. 188. Mean
  189. 189. average of the middle two numbers
  190. 190. Mode
  191. 191. Values that appear the most often in a set of numbers.
  192. 192. Acute Angles
  193. 193. • Angle whose measure is between 0 and 90 degrees.
  194. 194. Obtuse Angles
  195. 195. • Angle whose measure is between 90 and 180 degrees.
  196. 196. Complementary Angles
  197. 197. • Two angles that sum to 90 degrees.
  198. 198. Right Angle
  199. 199. An angle that is 90 degrees
  200. 200. Supplementary Angles
  201. 201. • Two angles that sum to 180 degrees.
  202. 202. Straight Angle
  203. 203. • An angle that’s measure is 180 degrees
  204. 204. Vertical Angles
  205. 205. • 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.
  206. 206. Transversal
  207. 207. • A line that crosses two lines (they do not have to be parallel) creating special types of angles
  208. 208. Corresponding Angles
  209. 209. • 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.
  210. 210. Alternate Interior Angles
  211. 211. • 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.
  212. 212. Alternate Exterior Angles
  213. 213. • 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.
  214. 214. Same Side Interior Angles
  215. 215. • 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.
  216. 216. Same Side Exterior Angles
  217. 217. • 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.

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