End Behaviors/Naming Polynomials!! =)

Linear Equations: 

y= mx+b 

1 degree

0 turns 

Domain - x values
Range - y values referred to as f(x)

When m is Positive: 

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

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

When m is Negative 

domain → -∞, range → +∞ (rises on the left)

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

Quadratic Equations (parabolic equation)

y=ax² 

2 degree 

1 turn

(a+b)(c+d)

            

When a is Positive 

When a is Negative 

Identifying special situations in factoring

• Difference of two squares
  • a²- b² = (a + b)(a - b)
    • (x + 9)(x − 9)
    • (6x − 1)(6x + 1)
    • (x³ − 8)(x³ + 8)
• Trinomial perfect squares
  • a² + 2ab + b²= (a + b)(a + b) or (a + b)²
    • x²-4x+4
    • 16x² - 8xy + y² = (4x - y)²
    • x²+6x+9
  • ^{a2 - 2ab + b2 = (a - b)(a - b) or (a - b)2}
• Difference of two cubes
• a³ - b³
• 3 - cube root 'em
• 2 - square 'em
• 1 - multiply and change
  
• 
  
  Sum of two cubes
• • a³ + b³ 
  • • 3 - cube root 'em
    • 2 - square 'em
    • 1 - multiply and change
• Binomial expansion

Quadratic equations: Circles, Parabola, Ellipse, Hyperbola

Quadtratic Equations:

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

If there is an equation like 4x² + 4y² = 36 then it is a circle  because a=c. The a is 4 and the c is 4

If there is an equation like 2x² + 4y = 3 then it is a parabola because a or c equals 0

If there is an equation like 4x² - 4y² = 12 then it is a hyperbola  because a and c have different signs.

If there is an equation like 4x² + 3y² = 25 then it is an ellipse because a is not equal to c, and the signs are the same.






--------This is an ellipse--------

Multiplying Matrices

There are 3 steps to multplying matrices.

1) Write a dimension Statement

2) Row x column and Sum x products

3) Repeat until complete

For this picture, you would have to repeat this process for the 2nd row x 4th column

Naming Polynomials:

--Number of turns is always 1 less than the degree.--

Degree:

0- Constant 

1- Linear

2- Quadratic 

3- Cubic

4- Quartic

5- Quintic 

6 to ∞- nth Degree 

Terms:

Monomial 

Binomial 

Trinomial 

Quadrinomial 

Polynomial 

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

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

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

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

Dimensions of a Matrix

•    A matrix is a rectangular array of data. We would call it a table in using everyday terms, but it is not a table but a bracket.
• The dimensions are listed as row x column, or n x m, where n represents the number of rows and m represents the number of columns.

EXAMPLES:

Error Analysis

The slope is 10/5 or 2. Not 10/1. Therefore the equation is wrong.

 the point, (1, -2) works for the first equation,but not the second. And it must work in both equations in order for it to be a solution.





#20-the line should be a dotted line
#21-shading should be below the line

Graphing Absolute Value Functions

  Graphing: y = a|x-h|=k
 When graphing an absolute value equation, your vertex will be (h,k).

*Vertical translations are shown by k:

*Horozontal translations are shown by h: it moves in the opposite direction of the indicated sign. if h is positive, then the graph moves left. if h is negative, the graph moves right.

*Vertical stretch of shrink is shown by a number in front of the absolute value signs.

Systems of Linear Equations

There are three types of solutions.
-Consistent:  

*independent: one solution: a point on a graph where two lines intersect. These lines have two different slopes.*dependent: same slope, same y-intercept.

These equations are the same line. The solution is written  as "All #s"-Inconsistent:

 

 Two lines with the same slope, but different y-intercepts. These lines are parallel.