End Behaviors

domain approaches positive infinity, range approaches positive infinity (rises right)
domain approaches negative infinity, range approaches negative infinity (falls left)

domain approaches negative infinity, range approaches positive infinity (rises left)

domain approaches positive infinity, range approaches negative infinity (falls left)

domain approaches positive infinity, range approaches negative infinity (falls right)

domain approaches negative infinity, range approaches negative infinity (falls left)

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

Terms

monomial

binomial

trinomial

quadrinomial

polynomial

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}
• 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
• Binomial expansion
• (a + b)³ = (a+b)(a+b)(a+b)
• (a + b)⁴ = (a+b)(a+b)(a+b)(a+b)

Quadratic Functions

Standform: ax^2 + bx + cy^2 + dy + e=0
An equation where A=C, it a circle.
Example: 2x^2 +2y^2 = 16



An equation where A=C=0 is an example: 4x^2 + 8y +6

An equation where A or C has opposite signs is a
hyperbole
Example: 2x^2 - 2x^2 = 8


An equation where A=C and Signs are the same, is an ellipse.
Example: 3x^2 + 4y^2 = 16

Multiplying Matrices

   A       B
[3 2]  [2 4 5]
[2 4]*[3 1 4]
[2 1] 


Above are our matrices that I will use to demonstrate how to multiply matrices.
First, you must find the dimensions of the two matrices. Matrix A's dimensions are 3 by 2. Matrix B's dimensions are 2 by 3. In order to multiply, you must make sure the column mathces with the row. in this one, 2 X 3 X 3 X 2. this means we can multiply.

Second, we multiply.
Our answer should be:
[12 14 23]
[16 12 26]
[7    9  14]

Error Analysis

For this error, the student correctly used his solution and tested the problem. The first problem was correct. The student's mistake is that he did not use his solution for the second problem, which proves that his solution is incorrect.

For this problem, the student's equation is incorrect. X is going up by 5, so the slope should be 2X instead of 10X. the way I figured this out is because when I input my  X value into the equation that supposedly represents the table, I did not get the y value. With the new equation, y = 9 + 2X, the table will be correct.

For number 22, the person did not use the correct line shading. Instead of a full line, the line should be dashed to represent the less than instead of the less than/equal to. For number 23, the line is correct, but the shading is incorrect. instead of shading below, it should be shaded above due to the fact that Y is greater than the equation.

For number 20, the shading is correct, but the line should be dashed instead of full. For number 21, the line is correct, but the shading should be below instead of above, due to the fact that y is less than/equal to the equation.

How to graph absolute value equations.

y = a | x - h | + k

vertex: (h,k)

• A tells you whether the V opens up or down
• Similar to slope except you go up and right, then up then left, or down and right, then down and left.
• H moves the V right or left ( opposite)
• K moves the V up and down

System of Equations

Consistent Independent - Has 1 solution. Two different equations with different slopes. Lines cross at one point. Only Solution is where the lines cross.

Consistent Dependent - Has infinite solutions. Has many lines with the same slope and same y-intercept and the lines overlap themselves, therefore it is the same line.

Inconsistent - Has no solution. Has two or more lines with the same slope but different y-intercepts.

Dimensions of a Matrix

This Matrix is 1 X 3. I found this out because matrices go row by column. there is only one row and three columns.

This Matrix is 3 X 3. I know this because this matrix has 3 rows and 3  columns.



This Matrix is 3 X 2. I know this because the matrix has 3 rows and 2 columns