Transcript Review Topic 2

IB Math Studies – Topic 2
IB Course Guide Description
IB Course Guide Description
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.
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}
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}
AB=
A∩B=
and
B = {2, 4, 6}
Number Sets
Reals
Rationals
(fractions; decimals that
repeat or terminate)
Integers
(…, -2, -1, 0, 1, 2, …)
(0, 1, 2, …)
* (1, 2, …)
(no fractions;
decimals that
don’t repeat or
terminate)
 , 2, etc.
Natural
Counting
Irrationals
+
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.
Arithmetic Sequences
Arithmetic Sequences
Arithmetic Series
Geometric Sequences
Geometric Sequences
Geometric Series
Solving Pairs of Linear Equations
Solving a System of Equations
a.k.a. “simultaneous equations”
Substitution
Elimination
1) Solve one of the equations for one
of the variables.
1) Choose a variable to eliminate
2) Substitute into the other equation
2) Make coefficients opposite
numbers by multiplying
3) Solve
4) Substitute to solve for the
remaining variable.
3) Add the equations; solve.
4) Substitute to solve for the
remaining variable.
Or use GDC – Graph both Equations and find Intersection
Solve by Substitution or Elimination
3x – 2y = -3
3x + y = 3
x + y = 14
x–y=4
3x + 2y = 2
3x + y = 7
2x + y = 9
x + 4y = 1
4x – 5y = 3
3x + 2y = -15
Solving Quadratic Equations - Factoring
• Always look for _____
GCF first.
difference of squares
• Two terms usually means ________________
• Three terms usually means ______________
factoring trinomials
– x2 + bx + c  normal
– ax2 + bx + c  Hoffman Method
• Check your answer by __________.
multiplying
FACTOR
1) 3x2 + 15x
= 3x(x + 5)
2)
12x – 4x2
= 4x(3 – x)
3) (x – 1)2 – 3(x – 1)
= (x – 1)(x – 4)
4)
(x + 1)2 + 2(x + 1)
= (x + 1)(x + 3)
FACTOR
5) 9x2 – 64
= (3x – 8)(3x + 8)
6) 100a2 – 49
= (10a + 7)(10a – 7)
7) 36 – t10
= (6 – t5)(6 + t5)
8) a2b4 – c6d8
= (ab2 – c3d4)(ab2 + c3d4)
9) a4 – 81b4
= (a2 + 9b2)(a – 3b)(a + 3b)
FACTOR
10) w2 – 6w – 16
= (w – 8)(w + 2)
11) u2 + 18u + 80
= (u + 8)(u + 10)
12) x2 – 17x – 38
= (x – 19)(x + 2)
13) y2 + y – 72
= (y + 9)(y – 8)
14) h2 – 17h + 66
= (h – 11)(h – 6)
15) t2 + 20t + 36
= (t + 18)(t + 2)
16) q2 – 15qr + 54r2
= (q – 9r)(q – 6r)
17) w2 – 12wx + 27x2
= (w – 9x)(w – 3x)
18) 10 + 3x – x2
FACTOR
= (5 – x)(2 + x)
19) 32 – 14m – m2
= (16 + m)(2 – m)
20) x4 + 13x2 + 42
= (x2 + 7)(x2 + 6)
21) 5m2 + 17m + 6
= (m + 3)(5m + 2)
22) 8m2 – 5m – 3
= (8m + 3)(m – 1)
23) 4y2 – y – 3
= (y – 1)(4y + 3)
24) 4c2 + 4c – 3
= (2c + 3)(2c – 1)
25) 6m4 + 11m2 + 3
= (2m2 + 3)(3m2 + 1)
26) 4 + 12q + 9q2
= (2 + 3q)2
27) 6x2 + 71xy – 12y2
= (6x – y)(x + 12y)
FACTOR Completely
28) 24x2 – 76x + 40
= 4(2x – 5)(3x – 2)
29) 3a3 + 12a2 – 63a
= 3a(a + 7)(a – 3)
30) x3 – 8x2 + 15x
= x(x – 5)(x – 3)
31) 18x3 – 8x
= 2x(3x – 2)(3x + 2)
32) 5y5 + 135y2
= 5y2(y + 3)(y2 – 3y + 9)
33) 2r3 + 250
= 2(r + 5)(r2 – 5r + 25)
34) 3m2 – 3n2
= 3(m + n)(m – n)
35) 2x2 – 12x + 18
= 2(x – 3)2
Solving Quadratic Equations – Quadratic Formula