Sequeneces and Series

A sequence is an ordered collection of objects where repititon is allowed. The general form of  a

sequence is A1, A2, A3...An, An+1. There is the arithmetic sequence and the geometric sequence. A Series is the sun of the terms in a sequence. For this one as well, there is an arithmetic describes and a geometric series. Both series and sequences are used to prove or disprove patterns.

Graphing systems of Inequalities

There are three different graphs that one could stumble across when dealing with systems of inequalities. The equation of a line is y=mx+b, a parabola y=(x-h)^2+k, and the circle x^2+y=r^2. From one of these equations then, you have to pick a test point. Finally, you have to shade the plane containing the test point. If the test point satisfies the equations. If the sign of the inequality equation is greater than or equal to you, then the line of the graph would be solide. However, if the equation is less than one, then the line of the graph will be dotted. 

Cramer's Rule

Cramer's Rule is used for a system of linear equations that has as many equations as unknowns and is valid whenever the system has a unique solution. The main equations are x=DX/D and y=DY/D. The way the actual solving process is set up is you put only the coefficients of x and y then cross multiplying as you do in matrices in order to get an answer. Then for the second equation the x values would be replaced by the value of the sum and the same for the y too. The point of Cramer's Rule is to not need more than one variable in order to solve a system.

Systems of equations

There are two kinds of systems of equations, the inconsistent and consistent. If the system is inconsistent, then there is no solution and there are parallel lines. However, consistent mens that there is one solution which is called the independent and infinite solutions (dependent).

There are two methods in which systems of equations can be solved. The first is substitution and in substitution, you have t o first solve 1 equation for one variable then substitute it into another then solve for the variable. From this point you would have to back substitute in order to find the 2nd variable. The second method is elimination where you would interchange any 2 equations then multiply by a constant number then add one equation to the other to eliminate a variable. 

Graphs of Polar Equations

So when graphing polar equations, always begin by creating a Table-of-Values. If r=acos@, then the circle is along the x-axis with diameter "a". If r=asin@, then they exist with a diameter and if r=a then the circle is at (0,0) with radius "a". There are a couple different kinds of polar equations. First are Limacon which are basic circles that has the chess  part. The second are Cardiods, which are special cases of limacons. Lastly, rose curves and lemisactes are also polar equations.

Polar Coordinates

Polar Coordinates are basically the position of a point in a plane, the first being the length of a straight line represented by "r" which is connects the point to the origin, and the second angle "theta" (represented as @ in this blog post due to the fact I do not have a theta sign) is made by this line with a fixed line. There are four ways to get to the same point on a polar coordinate: (r,@), (-r,@+pi), (r,-2pi-@), (-r, -@-pi). In order to convert between rectangular (x,y) and polar (r,@), remember that the two are equal to each other therefore can be used in stead for each other. Remember these two key equations!

r= /x^2+y^2    (the square root of x squared plus y squared

and 

@=tan^-1(y/x)   (the inverse tan of y divided by x.

Rotating Conic Sections

In order to rotate a conic section, one must begin at the starting point which has the equation: Ax^2+Bxy+Cy^2+Dx+Ey+F=0.

The first step is to find the angle through the equation cot20=(A-C)/B where 0<theta<90.

The second step would be to replace the x's and y's in order to get x=x*cos(theta)-y*sin(theta) and y=x*sin(theta)+y*cos(theta).

The last step would be to use algebra to simplify the equation. You should know your basic trig functions in order to aid in the process of simplifying.