Identifying special situations in factoring

• Difference of two squares
• a²- b² = (a + b)(a - b)
• x^2 - 5^2= (x+5)(x-5)
• 3^2 - 4^2= (3+4)(3-4)
• 6^2 - y^2=(6+y)(6-y)

Trinomial perfect squares

• a² + 2ab + b² = (a + b)(a + b) or (a + b)²
• x^2+ 10x + 25=(x+ 5)(x + 5) or (x + 5)²
• 2^2+ 16x + 16=(2+ 4)(2 + 4) or (2 + 4)²
• y^2+ 12y + 36=(y+ 6)(y + 6) or (y + 6)²
• ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
• x^2- 10x + 25=(x- 5)(x - 5) or (x - 5)²
• 2^2- 16x + 16=(2- 4)(2 - 4) or (2 - 4)²
• y^2- 12y + 36=(y- 6)(y - 6) or (y - 6)²
• Difference of two cubes
  • a³ - b³
    • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
      • 8x³ + 27 = (2x)³ + (3)³ = (2x + 3) (4x² - 6x + 9)
      • 
        
• Sum of two cubes
  • a³ + b³ 
    • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
      • 3 examples
• Binomial expansion
• (a + b)³ = (a+b)(a+b)(a+b)
• (a + b)⁴ = (a+b)(a+b)(a+b)(a+b)

End Behaviors

Domain - x values
Range - y values

domain → +∞, range → +∞ (means it rises on the right)

domain → -∞, range → -∞ (means it falls on the left)

• domain → -∞, range → +∞ (means it rises on the left)
• domain → +∞, range → -∞ (means it falls on the right)

• domain → +∞, range → -∞ (falls on the right)
• domain → -∞, range → -∞ (falls on the left)

Degree
0- constant
1-linear
2-quadratic
3-cubic
4-quantic
5-quintic

Terms
monomial
binomial
trinomial
quadrinomial
polynomial

Quadratic Functions

How to identify quadratic functions:
Standardform: ax² + bx + cy² + dy + e= 0

If you have an equation like 4x² + 4y²=36  The equation is a circle, because a=c
Example of a circle

If a or c equals 0 the equation is a parabola (for example: 2x² + 4y= 3)
Example of a parabola:

If a or c have different signs the equation is a hyperbola ( for example: 4x² - 4y²= 12)
Example of a hyperbola

If you have an equation like 4x² + 3y²= 25 the equations is an ellipse, because a is not equal to c and the signs are the same
Example of an ellipse

Matrice multiplication

When you want to multiply matrices you have to do row*column and sum the products!
You will repeat this several times:

Chapter 2: Dimensions of a matrix

The number of rows and columns in a matrix are written in the form rows×columns.

If you have a matrix like this   __    __     
                                            / 1 2 3 /
                                            / 4 5 6 /
                                            / 7 8 9 /

it has 3 rows and 3 columns, so the dimensions are 3×3. You read it aloud " three by three"
Matrices like that are called "square matrix"

Always remember: rows×columns       

 This is is a 1×3 matrix                            

This is a 3×3 matrix

This is a 3×2 matrix

This is also a 3×3 matrix (square matrix!)

The Identity Matrix

This is an example for an identity matrix.

Error Analysis

Mistake1:  The mistake is that "x" is going up by 5 and not by 1. The slope would be 10/5. This equals 2 and not 10/1!

Mistake2: The mistake is that he only checked 1 equation, but he had to check both!

Mistake3:  The mistake of the first graph is that it is a solid line and not a dotted line!
The mistake of the second graph is that it is shaded below and not above!

Mistake4:
The mistake in the first graph is the solid line. It should be a dotted line.
The mistake in the second graph is that it is shaded above. It should be below.

Graphing y=a/x-h/+k (Absolute Value Functions)

When you want to graph a equations like y=a/x-h/+k you have to follow this rules:

1. The vertex on your graph will be (H,K)  

2. a tells you wether the V is opened up or down
    If its negative its opened down, if its positive its opened up


3. the h moves the V to the right or to  the left
    If the h is positive it moves to the left, if its negative it moves to the right


4. k moves the V up or down


This is how it looks like

If y is greater you shade above, when its less you have to shade below and use a dotted line!