Identiying Quadratic Equations

If a=c then the equation is a circle: 2x² + 2y²= 36

If  a or c equals 0 then the equation is a parabola: x² + 2y= 6

If a and c have different signs then the equation is a hyberbola: 3x² - 3y²= 12

If  a is not equal to c but the signs are the same then the equation is an ellipse: 3x² + 2y²= 36

Muliplying Matrices

Before you can begin to multiply matrices you need determine whether or not the matrices can be multiplied. To do that you must write a dimension statement.

[3 4] [1 2 3]
[5 6] [4 5 6]
[7 8]
The dimension statement for this would be 3X2 times 2X3.

• The numbers on the inside (in red) must be the same to carryout the multiplication process
• The numbers on the outside (in blue) tell what the dimension of the solution will be. In this particular problem, the solution will be a 3 X 3 matrix.

If the two matrices can be multiplied, you then multiply row X  column and add the sum of the products to get the solution to the problem.

Dimensions of a Matrix

• When counting matrices, you must count row X column.
• If a matrix has the same amount of rows as it does columns, then it is a square matrix/
• If a matrix has zeros with ones going down diagonally, it is an identity matrix.

This matrix has one row and three columns. Its dimensions are 1 X 3.

This matrix has three rows and two columns. Its dimensions are 3 X 2.

This matrix has three rows and three columns. It is a square matrix and also an identity matrix. Its dimensions are 3 X 3.

This matrix has three rows and three columns. It is a square matrix. Its dimensions are 3 X 3.

• For this particular problem the value of x is going up by 5, so the slope should be 10/5 or 2 and not 10/1. By inserting the points, the final equation should be solved. With this answer, y is not equal to 9+10x in the t-chart.

• In order for a particular point to be the solution to a system, it has to solve both equations. So in this case (1,-2) solves the first equation, but not the second equation of the system.

• For problem #22 the shading is correct, but the line should be a dotted line, not a soild line. For problem #23 the solid line is correct, but the shading should be above the line, not below it.

• For problem #20 the shading is correct, but the line should be a dotted line and not a solid line. For problem #21 the sloid line is correct, but the shading should be below the line, not above it.

Graphing Absolute Value Equations

• The formula for graphing absolute value equations is y=a|x-h+k|. The point (h,k) is the vertex, point a determines how high or low on the y-axis the sytem will be, and point k also determines how far left or right on the x-axis the system will be.
• Point k: One thing to know about moving the system to the left or the right is that if point k is negative then you move to right. If point k is positive then you move to the left. The reason for this is because "standard form" is negative, which in turn makes +h negative.
• Point a: If point a is positive then the system moves up and if point a is negative, it moves down.
• If the equation has a slope of 1 and starts on point (0,0), it can be written simply as y=|x|.
• The higher the slope of the equation, the smaller the width of the system is.
• If the slope is a fraction, then the width of the system increases.

Systems of Equations

Consistent independent equations have one solution (x, y), which is where the two lines intersect. Consistent dependent equations have infinitely many solutions because they are the same line and inconsistent equations have no solution because they are parellel.