Conic Sections Review

Parabolas:

The basic equations for a parabola are (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h).

P is the distance between the vertex and its focus, which is in the same direction as the parabola, and between the vertex and the directrix, or a line directly behind the parabola. (h,k) are the rectangular coordinates for the vertex.

Ellipses:

The basic equations for ellipses are:

[(y-k)^2]/a^2 + [(x-h)^2]/b^2              AND          [(x-h)^2]/a^2 + [(y-k)^2]/b^2       (h,k) are the rectangular coordinates of the ellipse's center. a is half of the major axis (longer "diameter") and b is half of the minor axis (shorter "diameter"). Hyperbolas:  Once again, (h,k) are the coordinates of the center. A and b are the distances from the center to the edge of a rectangle through which the asymptotes of the graph pass diagonally. A will also be the distance from the center to the vertices of the hyperbola

Rotating Conics Review

In this section, we learned how to find the equation and sketch the graph of a conic section that is not perfectly vertical or horizontal, but on tilted axes. This picture shows the three basic equations needed in this chapter:


Steps:

First, you must use the top equation to find the angle of rotation of the axes. You then use that angle to simplify the bottom two equations. The ultimate goal is to replace x and y in the original conic's equation with x' and y', or the new x and y that correspond with the new axes.

Parametric Equations

Parametric equations involve a third variable, usually t. This allows us to depict the passage of time on a graph. The first step is to eliminate the parameter, usually by substituting to equate the x and y variables. Then, graph the regular rectangular equation on a graph. Finally, insert a few values for t in the original equations and label them on the graph. Below are two example problems:

Summations and Limits to Find Area

The following picture shows the six equations used in this lesson:

Here is an example of a problem that utilizes nearly all of the above equations:

Equation #6 is used throughout the first half of this problem. The main strategy is to factor 1/n out of the equation, then putting it outside of the summation. Once 1/n is removed three times so it's 1/n^3, you have to split up the summation into three summations using rule #5. You then use rule #1, #2, and #3, respectively, to simplify each summation. Simplify, then take the limit.

Determinant of a Matrix

There are several strategies to finding the determinant of the matrix. The basic strategy for a 2X2 matrix is to multply the top left number by the bottom right number and subtract the product of the two remaining numbers. Anything bigger than a 3X3 requires a complicated process of finding minors and cofactors that can be seen in action here:

http://www.educreations.com/lesson/view/determinant-of-a-matrix/16850106/?s=ciem9G&ref=link

For 3X3's, you can rewrite the first two columns of the matrix to the right of the matrix and multiply numbers diagonally, then add and subtract. This strategy is also seen in that Educreations video.

Verifying Trigonometric Identities

This section features many problems in which a combination of trig functions must be changed to look like another set of trig functions. These problems are not that difficult now, as we have memorized many trig identities. Strategies on these problems:

• Work with only one side of the equation (choose the more complicated side). 
• Try to convert everything to sines and cosines.
• Simplify complex parts of the problem ASAP using trig identities.

NBA Real Plus-Minus

ESPN just came out with a new advanced metric called Real Plus-Minus (RPM). It is intended to be a perfected form of the normal plus-minus stat, which measures the gains or losses of points while a given player is on the field. The issue with the original plus-minus stat is that it doesn't factor in the impact of the other players on the court at the same time as the given player, so mediocre players who play alongside superstars will often excel in this stat as much as their superior teammates.

Nothing has been revealed about the actual calculation of this stat yet, but it allegedly factors in numerous variables on both offense and defense to calculate the number of points a player contributes to or takes away from his team per 100 possessions. Although it's difficult to do anything to calculate using this metric because the algorithm is not given, we can use the already-calculated RPM numbers to compare and contrast different teams.

Comparison of the Oklahoma City Thunder and the Miami Heat based on the RPM of starters:

OKC:

Russell Westbrook: 4.09

Thabo Sefolosha: 0.37

Kevin Durant: 6.77

Serge Ibaka: 3.10

Kendrick Perkins: -3.19

Total: 11.14

MIA:

Mario Chalmers: 1.48

Dwyane Wade: 2.13

Shane Battier: 1.60

LeBron James: 8.11

Chris Bosh: 3.75

Total: 17.07

Interestingly, Nick Collison, a backup big-man for the Thunder, is the sixth-best player in the NBA according to this statistic with an RPM of 5.81. Collison is the same height as Kendrick Perkins, whose RPM is a lowly -3.19; if they inserted Collison into the starting lineup for Kendrick Perkins, their RPM total for starters would be increased to 20.14! 

Although this stat is new and certainly imperfect to some degree, it does bring the value of some underrated players to light. Maybe this switch would improve OKC and make them even better than the defending-champ Heat.

Source:
http://espn.go.com/nba/statistics/rpm/_/sort/RPM