10.3 Hyperbolas

Here are the equations for a hyperbola:

The first equation is used for horizontal hyperbolas, while the second is used for vertical hyperbolas.

Below is an example of a problem that asks you to sketch a hyperbola. Unless the coefficients of x^2 and y^2 are already one, the first step to these types of problems is completing the square. This is done on the second line of this problem. (Don't forget to add equal amounts to the other side of the equation!) to sketch the graph, mark the center (h,k), then mark points that are "a" distance away in the direction "a" corresponds to (in this case, x). Do the same for "b" (in this case, in the y direction) and make a box; the asymptotes, or lines the graph will never touch but approach infinitely, will run diagonally out of the corners of the box. Then draw the graph as shown, leaving from the vertex, or the "a" points you marked, and approaching the asymptotes.

10.2 Ellipses

This section focuses on graphing and finding equations for 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 coordinates of the center. "a" and "b" are the distance between vertices and the center; a is the longer of the two. Although "c" is not in the equation, it is important to know that c is the distance between the center and each focus. c2 = a2 – b2 Some problems will give you the equation of an ellipse and you have to find out information from it (center, foci, vertices, and eccentricity) and sketch it using that information. Other problems will give you hints and ask you to find the equation. This problem (#33) gives you a sketch and the vertices, and asks for the equation:

Works Cited:
http://www.purplemath.com/modules/ellipse.htm

The REAL True Shooting Percentage

So there's this stat that ESPN NBA analysts use all the time called true shooting percentage. It is intended to measure a player's shooting ability more accurately, as it accounts for free throws and three-pointers. However, when I saw the equation used to calculate it, I didn't understand how it accomplished its goal at all. Here's the equation:

True Shooting Percentage = Total points / [(FGA + (0.44 x FTA)]

In my research, I learned that the equation was created to account for the number of possessions a shot consumes. This helps explain the (.44 x FTA) part of the equation, as usually, the other team gets the ball after a pair of free throws, and some free throws come after a made shot on a three-point play, so such a free throw is a chance for points that doesn't even take up a possession. 0.44 was reasonable because it was slightly less than one-half, or .5, and therefore accounted for the possessions a free throw consumes fairly well. 

However, this means that "True Shooting Percentage" is a misleading name for this statistic; it is more a measure of efficiency per possession than per shot. So, I made my own, more reasonable True Shooting Percentage equation:

Laski True Shooting Percentage = Total Points / [FTA + (2 x FGA) + 3PA]

Or, more simply: LTS% = Points Scored / Total Possible Points 

This equation encompasses all types of shots accurately, as each three-point shot is worth three points,each regular field goal is worth two, and each free throw is worth one. The resulting percentage represents the percentage of points a player scored out of the possible number of points they could have scored if they never missed a shot of any kind.

Stephen Curry's Laski True Shooting Percentage so far this season:

1459/[276 + (2 x 1108) + 492] = 0.4889 --> 48.9%

Works Cited:

http://theoldnorthking.blogspot.com/2013/01/true-shooting-percentage.html

http://espn.go.com/nba/statistics/player/_/stat/scoring

10.1 Parabolas

The basic equations for a parabola are (x-h)^2 = 4p(y-k) and (y-k)^2 = 4p(x-h). As you can see, these two equations are very similar, but the variables are just rearranged. The first equation is used to represent parabolas that go up-and-down, while the latter represents parabolas that go side-to-side. The p is equal to the distance between the vertex of the parabola and it's focus or it's directrix. (The focus is a point in the same direction of the parabola, and the directrix is a line that the parabola goes away from). Finally, (h,k) is the vertex pf the parabola.

Passer Rating: Russell Wilson

These are the steps to finding Passer Rating. You must add together the answers of four equations, then multiply by 100 and divide by 6.

Total 1:

1. Divide a quarterback's completed passes by pass attempts.
2. Subtract 0.3.
3. Divide by 0.2 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 2:

1. Divide passing yards by pass attempts.
2. Subtract 3.
3. Divide by 4 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 3:

1. Divide touchdown passes by pass attempts.
2. Divide by 0.05 and record the total. The sum cannot be greater than 2.375 or less than zero.

Total 4:

1. Divide interceptions by pass attempts.
2. Subtract that number from 0.095.
3. Divide that product by 0.04 and record the total. The sum cannot be greater than 2.375 or less than zero.

Final Steps:

1. Add the four totals you recorded.
2. Multiply that total by 100.
3. Divide by 6.
4. The final number is the passer rating.

I used this to calculate Russell Wilson's Quarterback Rating in the Super Bowl:

(2.1+1.31+1.6+2.375)X(100/6)=123.08

Works Cited:

http://football.about.com/od/frequentlyaskedquestions/ht/How-To-Calculate-A-Quarterback-Rating.htm

9.8 and 9.9 - Exploring Data

These sections expand on the main ideas of mean, median, and mode that we've known for a long time. But now, Variance and Standard Deviation are added. The equations for these are given in the chapter, but most of the problems involve too many numbers to do in a reasonable amount of time, so it's a good decision to just use standard deviation and variance calculators online. The following example finds the mean, median, and mode of the set 3, 7, 14, 15, 21, 21.

Mode: 21

Median: 14.5

Mean: 13.5

9.7 Probability

Probability problems can seem really complicated, but are often a lot simpler than they appear. At the base, every problem is asking you the ratio of favorable outcomes to total circumstances 

(# of successes/# of possibilities). For example, if you want to get a coin flip to be heads, there is one successful outcome, but two total outcomes (heads and tails), so the probability of getting heads is 1/2. There are twelve face cards in a deck of 52 cards, so the probability of drawing a face card is 12/52 (3/13). Let's try an example:

14.) In a 52-card deck, one card is drawn. What are the chances the number is six or lower?

• Ace, 2, 3, 4, 5, and 6 are six or lower, and there are four of each of those, so 6X4=24.
• There are 52 total cards, so the probability is 24/52 (Simplified: 6/13).