Math Notes

June 16 2010
1st Quarter - Advanced Algebra
2nd and 3rd Quarter - Trigonometry
4th Quarter - Statistics

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June 17 2010
Language, Notation and Numbers in Mathematics

1. Natural Numbers - most basic numbers, denoted by N
   ex. {1,2,3,...}
   
   
2. Whole Numbers - natural numbers with zero(0), denoted by W
   ex. {0,1,2,3,...}
   
   
3. Integers - denoted by Z
   A. Positive - greater than zero
   ex. {1,2,3,...}
   
   B. Negative - less than zero
   ex. {-1,-2,-3,...}
   
   
4. Rational Numbers - fractions and mixed numbers, denoted by Q
   ex. {1 1/2,3/5,2/3,...}
   
   
5. Irrational Numbers - denoted by H
   
   
6. Real Numbers - all rational and irrational, denoted by R

Fig. 1
Notation
{} - braces, denotes set
… - ellipses, denotes pattern of continuous indefinitely
∈ - denotes element
⊆ - denotes subset
⊂ - denotes proper subset
∅ - null set or empty braces
, - to separate elements of the set

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June 18 2010
Definitions:

1. algebraic term - collection of factors that may include numbers, variables or expressions within parentheses
   
2. constant - a term that consists of non-variable number
   
3. variable - a symbol, commonly a letter, used to represent unknown quantity
   
4. coefficient - constant factor of a term
   
5. algebraic expression - a single term or a sum or difference of terms
   *Proper subset - all the elements of a set

Reading Notations:
Q={p/q|p, q ∈ Z;q≠0}
............⇓
Q is set of p over q such that p and q element of integers where q is not equal to zero
*| - such that
*; - where

June 21 2010
Relations - a correspondence between two sets
.............- can be represented using (A) Mapping Notation, (B) Bar Graph, (C) Ordered Pair, ...............and (D) Rectangular Coordinate System or Cartesian Plane

A. Mapping Notation
Fig. 2

B. Bar Graph
Fig. 3
* - denotes zero-n, used when it does not start from zero(ex. years)
*x - independent
_y - dependent

C. Ordered Pairs
__..(2008, 50) (2009, 100) (2010, 150) (2011, 225)

D. Cartesian Plane or Rectangular Coordinate System
__..-used in the same way

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June 22 2010
Function - a relation that pairs each element from the domain with exactly one element from the range
Ex. Determine whether a relation is a function.

A.
... Fig. 4

B.
... Fig. 5
..................FUNCTION

C. (1,2) (2,3) (3,4) (4,5)
.......X........Y
.......1........2
.......2........3
.......3........4
.......4........5
.....FUNCTION

D.
... Fig. 6.1
...*Vertical Line Test - a graph is a function if and only if every vertical line intersects
...the graph at most one point
... Fig. 6.2

Finding the value of function f(x) = 2x²-3x

1. ƒ(3) = 2(3)²-3(3)
   ƒ(3) = 18-9
   ƒ(3) = 9
   
2. ƒ(-2) = 2(-2)²-3(-2)
   *f(-2) ≠ ƒ•(-2)
   ƒ(-2) = 14
   
3. ƒ(-x) = 2(x)²-2(-x)
   ƒ(-x) = 2x²+3x
   
4. ƒ(x+1) = 2(x+1)²-3(x-1)
   ƒ(x+1) = 2(x²+2x+1)-3x-3_____*special product
   ƒ(x+1) = 2x²+x-1

June 23 2010
Domain of Function - largest set of all real numbers for which the value of f(x) is a real number

Ex. A. ƒ(m) = 3(m)+2
... D_f = R or m ∈ (-∞,∞) or m = {m|m ∈ R}
Ex. B. ƒ(x) = 3/x+4
... D_f = R-{4} or x ∈ (∞,-4) ∪(-4,∞) or x = {x|x≠4}
Ex. C. ƒ(x) = √2x+3
...Note: 2x+3 ≥ 0
.......... 2x ≥ -3
.......... x ≥ -3/2
... D_f = {x|x ≥ -3/2} or x ∈ [-3/2,∞)
Ex. D ƒ(x) = √4+3x
...Note: 4+3x ≥ 0
.......... 3x ≥ -4
.......... x ≥ -4/3
... D_f = {x|x ≥ -4/3} or x ∈ [-4/3, ∞)
Ex. E ƒ(x) = 3x²+5
... D_f = R
Ex. F ƒ(x) = 3/5+x
... D_f = R - {5}

*Don't mind the extra blue text. There was a glitch so I had to change the color. There's nothing that special about it.

June 24 2010
Operations of Function
ƒ(x) = 3x+4..........g(x) = 2x-3

1. The sum of ƒ+g is the function defined by ƒ(x)+g(x)
   ...(ƒ+g)(x) = ƒ(x)+g(x)
   _________ = (3x+4)+(2x-3)
   _________ = 5x+1
   D_f+g = R
   
2. The difference of ƒ-g is the function defined by ƒ(x)-g(x)
   ...(ƒ-g)(x)=ƒ(x)
   ________ = (3x+4)-(2x-3)
   ________ = x+7
   D_ƒ-g = R
   
3. The product of ƒ•g is the function defined by ƒ(x)•g(x)
   ...(ƒ•g)(x) = ƒ(x)•g(x)
   ___._____ = (3x+4)•(2x-3)
   _____.___ = 6x²-9x+8x-12
   ______.__ = 6x²-x-12
   D_ĥg = R
   
4. The quotient of ƒ/g is the function defined by ƒ(x)/g(x)
   ...(ƒ/g)(x) = ƒ(x)/g(x)
   _________= 3x+4/2x-3
   D_ƒ/g = R-{3/2}

June 29 2010
Analyzing Graph of Function
...A. Even/Odd
.......Even Function (p. 207)
..........-a function ƒ is an even function if and only if for each point (x,y) on
............the graph of ƒ, the point (-x,y) is also on the graph
..........Function Notation: ƒ(-x)=ƒ(x)
*Mirror image with respect to the y-axis.
Ex.
Fig. 7
Ex. Even function, continue.
Fig. 8.1
________________________⇓
Fig. 8.2
Note: When folding, fold on y-axis.
.......Odd Function
..........-a function is an odd function if and only if for each (x,y) on the graph of ƒ the point (-x,-y) is also on the graph
Ex.
Fig. 9
Fig. 10
*Mirror image with respect to the point of origin.
Note: When folding, -x-axis on +y-axis.
*
Fig. 11
____________NEITHER
It does not lie on the point of origin and is not a mirror image with respect to the y-axis.

...B. Increasing/Decreasing
.......Given and interval (I) that is a subset of the domain with x₁ and x₂ in I and x₂>x₁
.......Ex. [-3,0] x₁:-3 x₂:0
.......*a function is increasing at I if ƒ(x₂)>ƒx₁ for all x₁ and x₂ in I
.......*a function is decreasing at I if ƒ(x₂)<ƒ(x₁) for all x₁ and x₂ in I
.......*a function is constant at I if ƒ(x₂) = ƒ(x₁) for all x₁ and x₂ in I
Ex. Given:
I: [0,∞)
Fig. 12
...x...|..ƒ(x)/y
x₁: 1 | __1
x₂: 2 | __4

4>1 so the given function is INCREASING at the given interval
*ƒ(s):y axis and s:x axis no matter what letter is used.

July 1 2010
Given: I: (-∞,0]
_____ Fig. 12

....x....|..ƒ(x)/y
x₁: -2 | __4
x₂: -1 | __1
ƒ(x₂)<ƒ(x₁) so the function at the given interval is DECREASING
Note: If from left to right the graph goes down, it is decreasing.
____. If from left to right the graph goes up, it is increasing.
Given: I: (-∞,∞)
Fig. 13
The function CANNOT BE DETERMINED at the given I.
*Look at graph, use I to know which certain part to look at.
...C. Maximum and Minimum
.......Global maximum-absolute maximum
________________..-names the largest range value over the entire domain
.......Local maximum-relative maximum
________________.-names the largest range value over the specified interval
.......Global minimum-names the smallest range value over the entire domain
.......Local minimum-names the smallest range value over the specified interval
Ex.
Fig. 14
D_ƒ: [-7,6]
R_ƒ: [0,10]
Global Maximum: (0,10)
Global Minima: (6,0) and (-7,0)

Ex.
Fig. 15
D_ƒ: R (because of arrows)
R_ƒ: [0,∞)
Global Maximum: N/A (infinity)
Global Minimum: (0,0)

July 7 2010
Composition of Function
Given two functions ƒ and g, the composition of ƒ with g is defined by (ƒog)(x) = ƒ[g(x)]
Ex. ƒ(x) = 3x+1
___g(x) = 2x
_____ 1. (ƒog)(x) = 3(2x)+1
_______________.= 3(g(x))+1
_______________.= 6x+1
_____ 2. (goƒ)(x) = 2(f(x))
_______________ = 2(3x+1)
_______________ = 6x+2
Ex. ƒ(x) = 2x+x
___g(x) = x+1
_____ 1. (ƒog)(x) = 2(g(x))+(g(x))
_______________ = 2(x+1)+(x+1)
_______________ = 2x+2+x+1
_______________ = 3x+3
_____ 2. (goƒ)(x) = (ƒ(x))+1
_______________ = 2x+x+1
_______________ = 3x+1

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June 12 2010
Linear and Quadratic Functions
(p.74, 206 and 294)
General Linear Equation
Ax+B = 0
Standard Linear Equation
Ax+By = C
Forms and Formulas for Linear Equation

1. Slope Formula
   y = mx+b
   m:slope
   b:y-intercept
   Ex. Write the standard form of the equation of each line.
   __y = -7/5x+1
   __5(y+7/5x) = (1)(5)
   __7x+5y = 5
   ___x_|_y_
   ___0_|_1
   __5/7.|_0
   
   __7(0)+5y = 5
   __5y/5=5/5
   __y = 1
   __7x+5(0) = 5
   __7x/7 = 5/7
   __x = 5/7
   Ex. y = 3/2x+5
   __ 2(-3/2x+y) = (5)(2)
   __-3x+2y = 10
   _____x_|_y_
   _____0_|_5
   __10/-3.|.0
   
   __-3(0)+2y = 10
   __2y/2 = 10/2
   __y = 5
   __-3x+2(0) = 10
   __-3x/-3 = 10/-3
   __x = 10/-3
   *When the slope is positive, the function is increasing.
   *When the slope is negative, the function is decreasing.
   

June 13 2010
Linear Functions

B. Point-slope Form
__.y-y₁ = m(x-x₁)
__.Ex. A. ♥P(0,5) ♦Q(3,2)
__. ♥: y-5 = 5-2/0-3(x-0)
__. y-5 = -1(x-0)
__. ♦: y-2 = -1(x-3)
__. Standard Form
__. ♥: y-5 = -1(x-0)
__. y-5 = -x
__. y+x = 5
__. ♦: y-2 = -1(x-3)
__. y-2 = -x+3
__. y+x = 5
__.Ex. B. ♥ C(3,5) ♦ S(6,8 )
__. ♥: y-5 = 8-5/6-3(x-3)
__. y-5 = 1(x-3)
__. ♦: y-8 = 1(x-6)
__. Standard Form
__. ♥: y-5 = 1(x-3)
__. y-x = 2
__. ♦: y-8 = 1(x-6)
__. y-x = 2
__. x and y intercept
__. y-x = 2
__. y-(0) = 2
__. y = 2
__. (0)-x = 2
__. x = -2

July 19 2010
Quadratic Function and its Application
*A quadratic function is in the form of ƒ(x) = ax²+bx+c
*The domain of quadratic function is all R, D_ƒ = {x|x∈R}
*Range is to be solved
a>0 Fig. 16.1
a<0 Fig. 16.2

Ex. A car manufacturer can produce 15 cars per month. The profit made from the sales of these cars is modeled by p(x) = -0.2x²+4x-3. Where p(x) is the profit in hundred thousand dollars per month and x is the number of cars sold. Based on this model.
a. Find the y-intercept and explain what it means
__p(0) = 0+0-3 = -3
__y-intercept: (0,-3)
__They loose 300,000 dollars if they can't sell the cars
b. How many cars should be made and sold to maximize profit.
__Find vertex(x) = -b/2a = 4/0.4
__∴ 10
c. What is the maximum profit?
__p(10) = -0.2(10)²+4(10)-3
__p(10) = 17
__∴ maximum profit = 1,700,000 dollars

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July 20 2010
Axis of Symmetry - vertical line passing through the vertex
*Completing the Square(CS) - (b/2)²
*Quadratic Formula - x=-b±√4ac-b²/2a
*The rest are in the problem set.

July 21 2010
Transformation
Basic Quadratic Formula
ƒ(x) = x² → opens upward
Fig. 17
g(x) = (x-1)² → ƒ(x) = a(x-h)²
h = 1 (→)
k = -2 (↓)
Ex. m(x) = x²+9x+4
a = 1 b = 9 c = 4
h = -9/2(1) = -9/2 = -4.5
k = 4(1)(4)=9²/4(1) = 16-81/4 = -65/4 = -16.25
ƒ(x) = -x¹ → opens downward

June 27 '10
Polynomial Functions
Long Division
A. (k³-k²-k-2)÷k-2

= k²+k+1

B. (-8x⁴+36x³+14x²+25x+25)÷x-5

= -8x³-4x²-6x-5

C. (r³+2r²-33r+7)÷r+7

= r² - 5r +2 - 7/ r + 7 or r² - 5r + 2, R: -7

D. (8v⁵+32⁴_5v+20)÷v+4

= 8v⁴ + 5

Synthetic Division
A.
Fig. 18
= k²+k+1
*2 is the negative coefficient of the divisor

B.
Fig 19
= -8x³-4x²-6x-5

C.
Fig. 20
= 8v⁴+5

D.
Fig. 21
= r²-5r+2-7/r+7

Factor Theorem
__ For a polynomial P(x)
___ 1. If p(c) = 0, then x-c is a factor of p(x)
___ 2. If x-c is a factor then p(c) = 0
_____A. p(2) = (2)³-(2)²-2-2
___________ = 8-4-2-2
___________ = 0
Remainder Theorem
__If a polynomial p(x) is divided by (x-c) using synthetic division the remainder is equal to p(c)

2nd Quarter
Quarter Outline:

1. Angle Measure, Special Triangles and Angles
   
2. Trigonometry of Right Triangle
   
3. Unit Circle
   
4. Trigonometry of Real Numbers
   
5. Graph of sine, cosine, secant, co-secant, tangent and cotangent
   
6. Fundamental Identities and Families of Identities
   
7. Constructing and Verifying Identities

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August 23 2010
Review Triangles and Properties of Triangles
Basic Properties

1. The sum of the interior angles is 180°
   
2. The combined length of any two sides exceeds that of the third.
   
3. Larger angles are opposite the larger side.

Similar triangles - all corresponding angles are congruent and corresponding sides are proportional
Fig. 1
b/f=c/e or b/c = f/e

Special Right Triangles
45-45-90

• The legs are equal
  
• The hypotenuse is √2 times the length of either leg

30-60-90

• The hypotenuse is 2 times the shorter leg
  
• The longer leg is √3 times the shorter leg

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August 24 2010
Angle Measure in Radians
*rotation - one side moves
Fig. 2
*associates angles with circles and rectangular coordinate system (Cartesian plane)
Fig. 3

Co-terminal angles-angles that share an initial and terminal side
Ex. Fig. 4
(+) co-terminal --> 360° + 60° = 420°
(-) co-terminal --> -360° + 60° = -300°

Definitions:
Standard Position - position of an angle when the initial side is at positive x-axis
Quadrantal Angle - angle at its standard position and the terminal side coincides with the axes
*All quadrantal angles are central angles
*All angles at standard position are central angles