Polar Graphs

Polar coordinates

Polar coordinates are a way to label a point on a plane using the radius and angle, rather than x and y coordinates such as (x,y). Some important terms include: Pole=origin, polar axis=ray from the pole, r=directed distance from O to P, and lastly, Theta=angle couterclockwise from polar axis to segment OP. Unlike normal (x.y) coordinates, polar coordinates can be different yet be at the same point! What do I mean? Well, since an angle can be measured counterclockwise or clockwise, a coordinate like (r,theta) will equal (-r, theta+pi).
You should know the ways to convert coordinates from polar to rectangular. The equations you need are as follows: 

Here is an example of a conversion: 

Graphs of Polar Equations

Polar Equations, as already learned, are different from rectangular equations and useful in their own ways. Polar coordinates correspond with Polar equations, which both use angles and radius. These equations can make much cooler shapes and figures that rectangular equations can not make due to its limitations. Some typical polar equations: 

Polar Equations can also have symmetry and be symmetrical to many axes. One can use the symmetry test to yield what the polar equation/graph is symmetrical to. The symmetry test is as follows: 

One should also use a graphing utility to check your graph or graph extremely complex equations/graphs. 

Ellipses and Hyperbolas Review

Ellipses and Hyperbolas are closely related in both equations and definitions, even though they look nothing alike, so they can be easily learned together. First of all, an ellipse is the set of all points (x, y) the sum of whose distances from two distinct fixed points (foci) is constant. Some terms include: vertices=The line through the foci that intersects the ellipse at two points, major axis=The chord joining the vertices, midpoint=center of the ellipse, minor axis=The chord that is perpendicular to the major axis. Ellipse equations: 

The Hyperbola is the set of all points (x, y) the difference of whose distances from two distinct fixed points (foci) is constant. The terms that were defined with the ellipse also apply to hyperbolas. Hyperbola Equations: 

As you can see, they are similar in many ways such as how their equations are presented. The main differences that can be confusing and need to be memorized are the differing signs in the equations. 

Vectors in Space Review

Vectors, can be associated with an animated villain or cool mathematical concept. Sadly, I will have to explain the math one. Vectors in space, unlike normal vectors, exist in a 3D field, and therefore have three coordinates that denote is position unlike vectors on a plane. A vector in space can be denoted by v= <v1,v2,v3>. This means that there are three axes in all, the x, the y, and the z.The zero vector is denoted by 0 = <0,0,0>. The unit vectors are as follows : i= <1,0,0>, j= <0,1,0>, and k= <0,0,1> in the direction of the positive z-axis. One can obtain the component form of a vector by subtracting the values of the coordinates of the starting point by the values of the coordinates of the terminal point. Equations for vectors in space are as follows: 

and an example of a vector in space: 

Polar Coordinates Review

Joel had taught the class about polar coordinates, a way to label a point on a plane using the radius and angle, rather than x and y coordinates. Some important terms include: Pole=origin, polar axis=ray from the pole, r=directed distance from O to P, and lastly, Theta=angle couterclockwise from polar axis to segment OP. Unlike normal (x.y) coordinates, polar coordinates can be different yet be at the same point! What do I mean? Well, since an angle can be measured counterclockwise or clockwise, a coordinate like (r,theta) will equal (-r, theta+pi).
You should know the ways to convert coordinates from polar to rectangular. The equations you need are as follows: 

Matrices and Systems of equations Review

Miss V's daughter, Kara, followed in her footsteps and taught the ways of Matrices and Systems of equations. The two ways to solve system of equations with matrices, Gaussian Elimination and Gauss-Jordan Elimination. To use these techniques, one must know Elementary Row Operations, the ways of which one can manipulate matrices to get numbers where they want. These operations include: 1) interchange any two rows 2) multiply a row by a non-zero constant 3) add one row to another. 

The Gaussian Elimination with Back-Substitution method steps are as follows:
-Write the augmented matrix of the system of linear equations
-Use elementary row operations to rewrite the augmented matrix in row-echelon form
-Write the system of linear equations corresponding to the matrix in row-echelon form, and use back-substitution to find the solution
Make sure that you have a leading 1 and that the main diagonal is consisted of 1's. Everything under the main diagonal has to be zeros but above the main diagonal there can be any numbers.

The Gauss-Jordan Elimination method steps are as follows:
-Write the augmented matrix of the system of linear equations
-Use elementary row operations to rewrite the augmented matrix in row-echelon form
-Write the system of linear equations corresponding to the matrix in row-echelon form
Make sure that you have a leading 1 and that the main diagonal is consisted of 1's. Everything under and above has to be zeros.