1. IB Math Studies – Topic 2

2. IB Course Guide Description

4. Set Language A set is a collection of numbers or objects. - If A = {1, 2, 3, 4, 5} then A is a set that contains those numbers. An element is a member of a set. - 1,2,3,4 and 5 are all elements of A. -  means ‘is an element of’ hence 4  A. -  means ‘is not an element of’ hence 7  A. -  means ‘the empty set’ or a set that contains no elements.

5. Subsets If P and Q are sets then: – P  Q means ‘P is a subset of Q’. – Therefore every element in P is also an element in Q. For Example: {1, 2, 3}  {1, 2, 3, 4, 5} or {a, c, e}  {a, b, c, d, e}

6. Union and Intersection P  Q is the union of sets P and Q meaning all elements which are in P or Q. P ∩ Q is the intersection of P and Q meaning all elements that are in both P and Q. A = {2, 3, 4, 5} and B = {2, 4, 6} A  B = A ∩ B =

7. Reals Rationals Integers (…, -2, -1, 0, 1, 2, …) Natural (0, 1, 2, …) Counting (1, 2, …) Irrationals Number Sets (fractions; decimals that repeat or terminate) (no fractions; decimals that don’t repeat or terminate) * +

8. Number Sets N* = {1, 2, 3, 4, …} is the set of all counting numbers. N = {0, 1, 2, 3, 4, …} is the set of all natural numbers. Z = {0, + 1, + 2, + 3, …} is the set of all integers. Z+ = {1, 2, 3, 4, …} is the set of all positive numbers. Z- = {-1, -2, -3, -4, …} is the set of all negative numbers. Q = { p / q where p and q are integers and q ≠ 0} is the set of all rational numbers. R = {real numbers} is the set of all real numbers. All numbers that can be placed on a number line.

9. Arithmetic Sequences

11. Arithmetic Series

12. Geometric Sequences

14. Geometric Series

15. System of Equations Solving a System of Equations a.k.a. “simultaneous equations” Substitution 1) Solve one of the equations for one of the variables. 2) Substitute into the other equation 3) Solve 4) Substitute to solve for the remaining variable. Elimination 1) Choose a variable to eliminate 2) Make coefficients opposite numbers by multiplying 3) Add the equations; solve. 4) Substitute to solve for the remaining variable. Solving Pairs of Linear Equations Or use GDC – Graph both Equations and find Intersection

16. Solve by Substitution or Elimination x + y = 14 x – y = 4 2x + y = 9 x + 4y = 1 3x – 2y = -3 3x + y = 3 3x + 2y = 2 3x + y = 7 4x – 5y = 3 3x + 2y = -15

17. Always look for _____ first. Two terms usually means ________________ Three terms usually means ______________ – x 2 + bx + c  normal – ax 2 + bx + c  Hoffman Method Check your answer by __________. Solving Quadratic Equations - Factoring GCF difference of squares factoring trinomials multiplying

18. FACTOR 1)3x 2 + 15x 2) 12x – 4x 2 3)(x – 1) 2 – 3(x – 1) 4) (x + 1) 2 + 2(x + 1) = (x – 1)(x – 4) = 3x(x + 5) = 4x(3 – x) = (x + 1)(x + 3)

19. FACTOR 5)9x 2 – 64 6)100a 2 – 49 7)36 – t 10 8) a 2 b 4 – c 6 d 8 9) a 4 – 81b 4 = (a 2 + 9b 2 )(a – 3b)(a + 3b) = (3x – 8)(3x + 8) = (10a + 7)(10a – 7) = (6 – t 5 )(6 + t 5 ) = (ab 2 – c 3 d 4 )(ab 2 + c 3 d 4 )

20. FACTOR 10) w 2 – 6w – 16 11) u 2 + 18u + 80 12) x 2 – 17x – 38 13) y 2 + y – 72 14) h 2 – 17h + 66 15) t 2 + 20t + 36 16) q 2 – 15qr + 54r 2 17) w 2 – 12wx + 27x 2 = (u + 8)(u + 10) = (x – 19)(x + 2) = (h – 11)(h – 6) = (t + 18)(t + 2) = (q – 9r)(q – 6r) = (w – 9x)(w – 3x) = (y + 9)(y – 8) = (w – 8)(w + 2)

21. FACTOR 18) 10 + 3x – x 2 19) 32 – 14m – m 2 20) x 4 + 13x 2 + 42 21) 5m 2 + 17m + 6 22) 8m 2 – 5m – 3 = (m + 3)(5m + 2) = (8m + 3)(m – 1) 23) 4y 2 – y – 3 24) 4c 2 + 4c – 3 25) 6m 4 + 11m 2 + 3 26) 4 + 12q + 9q 2 27) 6x 2 + 71xy – 12y 2 = (2 + 3q) 2 = (5 – x)(2 + x) = (16 + m)(2 – m) = (2m 2 + 3)(3m 2 + 1) = (x 2 + 7)(x 2 + 6) = (2c + 3)(2c – 1) = (y – 1)(4y + 3) = (6x – y)(x + 12y)

22. FACTOR Completely 28) 24x 2 – 76x + 40 29) 3a 3 + 12a 2 – 63a 30) x 3 – 8x 2 + 15x 31) 18x 3 – 8x = 2x(3x – 2)(3x + 2) 32) 5y 5 + 135y 2 33) 2r 3 + 250 34) 3m 2 – 3n 2 35) 2x 2 – 12x + 18 = 2(x – 3) 2 = 4(2x – 5)(3x – 2) = 3a(a + 7)(a – 3) = 3(m + n)(m – n)= x(x – 5)(x – 3) = 2(r + 5)(r 2 – 5r + 25) = 5y 2 (y + 3)(y 2 – 3y + 9)

23. Solving Quadratic Equations – Quadratic Formula