Naming Polynomials and End Behaviors

Degrees

-0 degree-constant

-1st degree-linear

-2nd degree- quadratic ( U- shaped )

-3rd degree-cubic ( S- shaped )

-4th degree- quartic

-5th degree-quintic

The number of turns a polynomial has is always one less of their degree.

Terms

monomial- one term
binomial- two terms
trinomial- three terms
quadrinomial- four terms
polynomial- more than four terms

Linear Equations
1st degree- 0 turns
y=mx+b

When the slope of a line is positive, then the line rises to the right and falls to the left.

When the slope of a line is negative, the the line rises to the left and falls to the right.

Quadratic Equations

2nd degree- 1 turn

y=ax² 

(a+b)(c+d)

When a parabolic graph is positive, then it rises right and left.

When a parabolic equation is negative then it falls both and left and right.

Identifying Quadratic Functions

The standard form of a quadratic function is  ax² + bx + cy² + dy + e= 0.

The function 4x²+4y²=36 is a circle because a=c (or the coefficients for a are equal to those to c).

The function 2x²+4y²=3 is an ellipse because a does not equal c.

If a or c equal zero, then the equation is a parabola, like 3x+5y²=6.

The function 7x-8y=4 is a hyperbola because a and c are different signs.

Multipying 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:

     2 x 2              2 x 2                                                                   

             

     

                                                                  

         2 x 2   times    2 x 2 

The numbers highlighted show that the matrices can be multiplied, since the inside numbers are the same.

  2 x 2   times    2 x 2

These numbers become the dimensions for the product matrix.

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.

Dimesions Of a Matrix

To find the dimensions of the matrix, you need to count how many columns and rows there are. 

The rows are the numbers going horizantally.

 

The columns are the numbers going vertically.

 For the example below, there are two rows and three columns. So the dimensions would be written 2 x 3. The dimension format will always be written ROW x COLUMN.

 For the matrix below, the dimensions are 3 x 3. This matrix would also be classified as a square matrix, since the number of rows and columns are the same.

The matrix below has the dimensions 3 x 3. This matrix is called an identity matrix, because the numbers going across the diagonal is 1.

Error Analysis

1)

The error in this equation is that the slope goes up by 2, not 10, so the equation should read y=2x+9

2)

The error with this system of equations is that the student should have plugged the solution into both  equations, not just one, to check and see if the point works for both problems.

3)

For the first inequality, the error is that the line should have been dotted, since the actual equation isn't included in the solution. For the second inequality, the solutions should have been shaded above the line, since the equation says "greater than or equal to".

4)

For the first inequality, the line should be dotted, again, since the actual equation isn't included in the solution. For the second inequality, the solutions should be shaded below the line since the equation reads "less than or equal too".

How to Graph Absolute Value Equations

The format for an absolute value equation is y=a|x-h|+k, where the vertex is (h,k).

• "a" in the equation determines weather the graph (which is shaped like a "V") opens up or down. For example, if "a" is negative, then the graph will open downwards (the "V" would be upside-down). "a" is similar to the slope except with "a"  you would go up and right, then up and left, or if "a" is negative, down and right and then down and left. You're basically reflecting the same points over the axis of symmetry (the line that goes through the middle of the graph).

• "h" moves the graph to the left or right opposite of the sign given. So, if the sign is positive, then the graph moves to the left.

• "k" moves the graph up or down corresponding with the sign. For example, if the sign is negative then the graph moves down.

  

How To Graph Absolute Value Equations

The equation for an absolute value graph is y=a|x-h|+k.

• "a" tells you weather the graph opens up or down (if "a" is negative, then the graph will open down). Its similar to the slope except

Systems of Equations

Consistent Independent Graph: Has one solution and has different slopes

 

   

Consistent Dependent Graph: All the numbers on both lines are solutions; the lines have the same slope and same y-intercept (SAME LINE!!!!) .

Inconsistent Graph: Doesn't have a solution; has the same slope but different y-intercepts (PARALLEL!!!)