Linear Equations and End Behaviors

Naming Polynomials: Degree/ Terms

- 0: Constant/ Monomial
- 1: Linear/ Binomial
- 2: Quadratic/ Trinomial
- 3: Cubic/ Quadrinomial
- 4: Quartic/ Polynomial

When the line falls to the left and rises to the right, m is positive.

When the line falls to the right and rises to the left, m is negative.



Types Of Factoring

The Difference Of Two Squares:


a²- b² = (a + b)(a - b)


Ex:

    1. (x+4)(x-4)
    2. (x+6)(x-6)
    3. (x-8)(x+8)


Trinomial perfect squares


a² + 2ab + b²= (a + b)(a + b) or (a + b)<sup>2


Ex:
 1.) (x+9)(x+9)
 2.) (x+5)(x+5)
 3.) (x+7)(x+7)</sup>



 


 Difference of two cube

Ex:

• 8x³ + 27 = (2x)³ + (3)³ = (2x + 3) (4x² + 6x + 9) 

* 1- Cube root em
* 2- Square root
* 3- multiply and change


Binomial expansion





a³ - b³




3 - cube root 'em
2 - square 'em
1 - multiply and change


3 examples


 Sum of two cubes


a³ + b³ 


3 - cube root 'em
2 - square 'em
1 - multiply and change

Quadratic Functions

*The standard form of a quadratic equation is: ax² + bx + cy² + dy + e= 0

If an equation looks something like  6x+5y²=25, then it's an ellipse because the signs are the same and a is not equal to c.

 

If the equation looks something like 3x²+5y²= 15, then it's a parabola  
because a or c is equal to 0.








If the equation looks something like

4x²-4y²=12, then the equation is a
hyperbola  because the a and c are the
same coefficient but with different 
signs.                      





If the equation looks something like 3x²+3y²=18, then the equation is a

circle because a is equal to c.                                                  

Multpliying Matrices

                                  



To multiply matrices, you multiply Row x Column 

  * In this case, you multiply the first row by the first column and the second row by the first column since there is only one column in the second matrix.

Dimensions of a Matrix

Dimensions of a matrix consists of rows and columns:

                          Matrix
                          [2 1]
                          [4 3]
Rows of a matrix go horizontally
                          Rows
                         [3 0 -7]
                         [-6 2 3]
Columns of a matrix go vertically
                      Columns
                       [2]
                                  [1]
                      [-9]

In order to find the dimension of a matrix you have to count the number of rows x columns:  

[ 3 1 6]: This is a 1x3 matrix because it has one row and 3 columns

[5  9]:This is a 2x2 matrix because there are two rows and 2 colmns 

[2 -4]


Identity Matrix:
 [0 1]: If you multiply the identity matrix with any other matrix
 [1 0]   then the product will be the non-identity matrix.

  [-3 1 ] x [0 1] =  [-3 1]
    [6   8]    [1  0]     [6 8]
     

Error Analysis

The error in this problem had to do with the equation.

The first problem was that it was not in slope-intercept form which would be y= 10x +9. The other problem is, the slope is incorrect. It should be 10/5or 2, making the quation y=2x+9 because the"y"s dont go up by ten. they go up by five.

Point (1,-2) is not a solution because it only solves the first equation. If u plug in 1 for x and -2 for y the piont will work in the first equation but not for the second one.

For problem #20, the error had to do with line. The line should be dotted and not solid because the equation does not say greater than and equal to or less than and equal to.

For problem #21, the problem was the position of the shading. Because the line is solid, the shading should be below the line and not above it.

For problem #22, the line that was graphed should be dotted and not solid because it doesn't say greator than and equal to or less than or equal to.
 For problem #23, the problem was with  the location of the shading. The shading should be above and not below the line.

Graphing Absolute Value Equations

Before graphing Y=a|x-h|+k, you have to know the importance of each variable:

• a: tells whether the graph opens up or down; if the coefficient in front of the a is negative then the graph opens down and if it's positive then it opens up.
• h: tells whether the graph moves left or right; if the h is negative then the graph moves to the right and if it's positive then it moves to the left
• k: tells whether the graph moves up or down

*The vertex of the graph is (h,k)
* If there is a negative in front of the absolute value then the graph   will be reflected over the x-axis
*Y=|x+h| moves "h" units to the left
*Y=|x-h| moves "h" units to the right
*Y=|x|+k moves "k" units to the left.
*Y=|x|-k moves "k" units to the right

Sysyems of Equations

Consitent-independent lines consist of different slopes, (x,y) which equals one solution. Consistent-dependent lines consist of the same line, same slope and y-intercepts and ALL REAL #s. Inconsistent lines are parallel with the same slope, different y-intercepts and NO SOLUTION.

Consistent-independent Line

Consistent-dependent

Inconsistent