End Behaviors

Domain - x values

Range - y values referred to as f(x)

 

 

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

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

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

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

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

More on Multiplying Matrices

Scalar multiplication is when you distribute the number outside the matrix to all the numbers inside the

brackets.



To multiply matrices, you first need to write a dimension statement. The dimension statement basically states that the columns of the first matrix must match the rows of the other matrix
For example:

After you determined that the matrices can be multiplied, then you start to multiply them together. To do this, you would multiply the first row of the first matrix with the column of the second matrix. More specifically, you would multiply the first number of the first row on the first matrix with the first number of the first column of the second matrix. You then add the products together and thats the first number of the product matrix. You repeat this until all the numbers of both the matrices have been multiplied, giving you your product matrix.

Dimensions of a Matrix and Multiplying Matrices

This is a matrix. A matrix has rows and columns.





Rows are horizontal...

and columns are vertical.

In order to write a dimensions statement, you gotta count the number of ROWS and then number of COLUMNS.

 This is a 3x3 matrix because it has 3 rows and 3 columns. Also known as a square matrix and the identity matrix. 



This is a 3x3 column because it has 3 rows and 3 columns. Also known as a square matrix because it has the same number of rows and columns.

This is a 1x3 matrix because it has 1 row and 3 columns.



This is a 2x2 matrix because it has 2 rows and 2 columns. Also known as a square matrix.

By knowing the dimensions statement, you can figure out if you can multiply two matrices or not. For example, you can multiply a 3x2 matrix and a 2x5 matrix because the two inside numbers are the same. The answer will be a 3x5 matrix because the two outside numbers always give you your answer.

  

Error Analysis

The problem with this is that the x-axis is not going up by increments of one. The correct answer is y = 2x + 9

The problem with this is the student used the solution to check only one equation. In order for the solution to be the correct answer, it MUST solve both equations, not just one

For number 22, the line is supposed to be dashed not solid.

For number 23, its supposed to be shaded up, not down.

 

For number 20, the line is supposed to be dashed, not solid.

For number 21, the shading should be below the line, not above it.

Graphing Absolute Values

y = a|x-h|+k

• 
  
  A negative in front of the a term flips it down.
• 
  
  When a is less than one, verticle compression. When a is greater than one, verticle stretch.
• 
  
  Negative between absolute value bars means move to the right. Positive between absolute value bars means move to the left.
• 
  
  K is the y-intercept.
• 
  
  The vertex is (h,k)

Systems of Equations

Consistent and Independent
- Has solutions
- Different slopes, different y-intercepts

Consistent and Dependent
- Same slope, same line

Inconsistent
- Same slope, different y-intercepts
- has no solutions