What’s on the NC Math 2 Final Exam?
𝑥 +𝑥
𝑦 +𝑦
Midpoint = ( 1 2 , 1 2 ) distance = √(𝑥1 − 𝑥2 )2 + (𝑦1 − 𝑦2 )2
2
2
Translation (slide)
Reflection (flip)
Rotation (spin)
𝑇𝑎,𝑏 (𝑥, 𝑦) = (𝑥 + 𝑎, 𝑦 + 𝑏)
𝑟𝑥−𝑎𝑥𝑖𝑠 = (𝑥, −𝑦)
𝑅900 (𝑥, 𝑦) = (−𝑦, 𝑥)
CCW
𝑟𝑦−𝑎𝑥𝑖𝑠 = (−𝑥, 𝑦)
𝑅1800 (𝑥, 𝑦) = (−𝑥, −𝑦) CCW
Dilation
𝑟𝑦=𝑥 = (𝑦, 𝑥)
𝑅2700 (𝑥, 𝑦) = (𝑦, −𝑥)
CCW
𝐷𝑘 = (𝑘𝑥, 𝑘𝑦)
𝑟𝑦=−𝑥 = (−𝑦, −𝑥)
𝑅−900 (𝑥, 𝑦) = (𝑦, −𝑥)
Clockwise
-preimage is the original points (𝑥, 𝑦) ; image is the new shape (𝑥 1 , 𝑦1 ) -multiple transformations!!!
-dilate from a given center and scale factor
-identify combinations of transformations
-line segments of preimage and image of dilation are PARALLEL
that graph a preimage to its image (& reverse)
Unit 1 Transformations
Unit 2 ∆ Angle Sum Theorem (all angles add up to 180⁰)
-median: connects midpoint of ∆ side to opposite corner
-centroid: intersection of all 3 medians
Centroid Theorem
-centroid is ⅔ distance of median from corner to midpoint
-∆ Inequality Theorem (sum of any 2 sides is > 3rd side)
-midsegment: connects 2 midpoints of sides of a ∆
Midsegment Theorem
-midsgmt is parallel AND exactly ½ length of 3rd side of ∆
Triangles: right, obtuse, acute, equilateral, isosceles, scalene
-Ext. Angles Theorem: ext. angle is = to sum of 2
Identify: line segments, rays, parallel lines, transversals
interior angles
Identify: interior, exterior, alternate, and consecutive angles
*base angles of isosceles ∆ are congruent!*
Angles: adjacent, vertical, linear pairs, complementary, supplementary, perpendicular…solve for angle measures!!!
Determine if ≅ or supplemental: corresponding, alternate interior, alternate exterior, consecutive interior angles
Unit 3 SSS: 3 sides of 2 ∆’s are ≅ SAS: 2 sides and the included angle are ≅ ASA: 2 angles and included side are ≅
AAS: 2 angles and the NON-included side are ≅
HL: hypotenuse and leg of 2 right ∆’s are ≅
*no AAA or ASS/SSA for triangle congruency!!
*included means “in between”
CPCTC: use it to prove that angles or sides of congruent triangles are congruent
Triangle Similarity: 2 triangles are SIMILAR if they have AA; all their sides have the same RATIO (think dilations)
Unit 4 Pythagorean Theorem and Converse
right: 𝑎2 + 𝑏 2 = 𝑐 2
acute 𝑎2 + 𝑏 2 > 𝑐 2
obtuse: 𝑎2 + 𝑏 2 < 𝑐 2
45 45 90 ∆
30 60 90 ∆
Trig Functions
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
sine = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
cosine = ℎ𝑦𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑒
tangent =
𝑜𝑝𝑝𝑜𝑠𝑖𝑡𝑒
𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡
-SOH CAH TOA for missing side
-use 𝑠𝑖𝑛−1, 𝑐𝑜𝑠 −1 , 𝑡𝑎𝑛−1 to find missing angle
*put calculator in degree mode!*
Unit 5 —FOIL; multiply 3 sets of binomials to create a quadratic expression ex: (𝑥 + 1)(𝑥 − 3)(𝑥 + 3)
-factor and solve: trinomials, difference of squares (ex: 𝑥 2 − 100 = 0), and GCF (ex: 6𝑥 2 + 18𝑥 = 0)
- factor and solve for 𝑥 with 𝑎 > 1 if 𝑦 = 𝑎𝑥 2 + 𝑏𝑥 + 𝑐 (use box method or X)
-Solve for 𝑥 using Quadratic Formula 𝑥 =
“𝑏 2 − 4𝑎𝑐” is the discriminant:
+ discriminant: two solutions
0 = discriminant: one solution
- discriminant: NO solution
−𝑏±√𝑏2 −4𝑎𝑐
2𝑎
What does 𝒂 tell us?
+ a: parabola opens UP
- a: parabola opens DOWN
𝑎 > 1: graph gets WIDER
𝑎 < 1: graph gets NARROWER
Graph in:
𝟐
Standard: 𝒚 = 𝒂𝒙 + 𝒃𝒙 + 𝒄
Factored: 𝒚 = 𝒂(𝒙 − 𝒑)(𝒙 − 𝒒)
Vertex: 𝒚 = 𝒂(𝒙 − 𝒉)𝟐 + 𝒌
(𝑝, 0) and (𝑞, 0): x-intercepts
(ℎ, 𝑘) is vertex
𝑐: y-intercept
𝑏
Axis of Sym: 𝑥 = − 2a
+ ℎ shift right
+ 𝑘 shift up
(𝑥 of the vertex; max/min)
- ℎ shift left
- 𝑘 shift down
Identify: vertex, axis of symmetry, x- and y- intercepts, max/minimum, domain and range based on graph/equation
-interpret a situational word problem and solve using the appropriate quadratic equation
-may need to find vertex, x- , y-intercepts, restricted domain, max/min, or a given point within the scenario
Unit 6
Ex: Solve by completing the square: 4x2 + 8x – 9 = 0
Completing the Square: change equations from standard for to vertex form; solve
by completing the square
Regression! Use the STAT feature of graphing calculator to create linear, quadratic,
or exponential equation
-type x and y values into 𝐿1 and 𝐿2 ; stat→calc; choose LinReg, QuadReg, ExpReg,
etc. Graph equation
-turn on Stat Plot 1; Zoom9 to get graph of best fit
-2nd 0; diagnosticON gets the 𝑟 2 value (1 is best fit)
-see calculator cheat sheet for tricks!
-use graph to find max/min, intercepts, etc.
Unit 7 -simplify complex algebraic expressions using rules of exponents (see half-sheet with rules of exponents)
5
7
-simplify radical expressions (ex: √72𝑥 5 𝑦 6 )
-rewrite radicals as rational exponents (and back again!) ex: 𝑥 7 = √𝑥 5
-graph Radical Equations: 𝒇(𝒙) = 𝒂√𝒙 − 𝒉 + 𝒌
-solve radical and
rational expressions for x
1
a: initial amount
+ ℎ shift right
- ℎ shift left
Domain: [h,∞)
Ex: 𝑥 + 32 − 1 = 𝑥 and
+ 𝑘 shift up
- 𝑘 shift down
Range: [k, ∞) or (-∞, k]
15
𝑥
=9
Variation
Inverse
Direct
𝑦 = 𝑘∗𝑥
𝑦=
𝑘
𝑥
Joint
𝑦 = 𝑘∗𝑥∗𝑧
Unit 8 -solve by GRAPHING and SUBSTITUTING non-linear systems of equations: 1 linear and 1 quadratic,
Ex: 𝑦 = 𝑥 2 + 4𝑥 − 5 and 𝑦 = 2𝑥 − 5
2 quadratics,
standard form (x-and y-intercepts): 𝐴𝑥 + 𝐵𝑦 = 𝐶
𝑦 −𝑦
slope = 𝑚 = 𝑥2 −𝑥1
𝑏: y-int.
𝑦 = 𝑚𝑥 + 𝑏
2
1
-write system of equations/inequalities based on word problem
-solve system of 2 inverse variations using substitution
−4
Ex: 𝑦 = 𝑥 + 4 and 𝑥 = 𝑦
2 radical equations (graph only )
Ex: 𝑦 = (𝑥 − 1)2 − 6 and 𝑦 = −𝑥 2 + 7
1 linear and one radical
Ex: 𝑦 = √𝑥 + 4 + 1 and 𝑦 = 3 − 𝑥
Ex: 𝑦 = √𝑥 − 2 − 3 and 𝑦 = −2√𝑥 − 2
Unit 9 —see formula sheet for probability formulas
Calculate basic probabilities: 𝑃(𝐸) =
# 𝑜𝑓 𝑝𝑜𝑠𝑠𝑖𝑏𝑙𝑒 𝑒𝑣𝑒𝑛𝑡
# 𝑜𝑓 𝑠𝑎𝑚𝑝𝑙𝑒 𝑠𝑝𝑎𝑐𝑒
Complements: everything EXCEPT the event (1- P(E) )
Union (OR)
E = A U B (add the events)
-M.E. (disjoint) or non-M.E. events
-set up and solve Venn Diagrams
-two-way tables
-conditional probability
Intersection (AND)
E = A ∩ B (multiply the events)
-independent or dependent events