Jordan Massoud Blog Post #2- Functions in Real Life

Jordan Massoud

Math-160-001

Blog #2

Part A:

1.     My source is a NASCAR standings data sheet posted midway through the season: http://nytimes.stats.com/nascar/index.asp?page=standings&series=NASCAR.

2.     A function is a relationship in which one output is paired with exactly one input.

3.     On the “Sports” page of the New York Times, I found a data chart listing the NASCAR Sprintcup Standings with related statistics listed for each driver:

Rank	Driver	Points	Behind	Starts	Wins	Top 5	Top 10	DNF	Laps	Laps Led
1	Brad Keselowski	3000	-	29	5	13	16	1	8220	1496
2	Jeff Gordon	3000	0	29	4	11	19	0	8289	665
3	Joey Logano	3000	0	29	4	13	18	4	8179	794
4	Jimmie Johnson	3000	0	29	3	10	18	3	7883	1035
5	Kevin Harvick	3000	0	29	2	10	15	2	7860	1592
6	Matt Kenseth	3000	0	29	0	11	18	2	8128	467
7	Denny Hamlin	3000	0	28	1	6	12	2	7495	195
8	Kyle Busch	3000	0	29	1	6	12	4	7741	411
9	Ryan Newman	3000	0	29	0	2	11	1	8264	24
10	Dale Earnhardt Jr.	3000	0	29	3	11	17	2	7667	220
11	Carl Edwards	3000	0	29	2	6	11	2	8246	134
12	Kasey Kahne	3000	0	29	1	3	10	3	8181	204
13	AJ Allmendinger	2077	923	29	1	2	4	3	8097	68
14	Kurt Busch	2073	927	29	1	6	8	5	8064	181
15	Greg Biffle	2072	928	29	0	3	10	1	8182	109
16	Aric Almirola	2061	939	29	1	2	7	6	7581	23

4.     There are many possible relationships present in this data chart. I am examining the relationship between the position in the standings (i.e., 1^st, 2nd, 3rd) and the number of laps the person in that position has led.  I am going to let “N” be the number of laps led, and “R” be the ranking in the standings. Represented in function notation, it would be: R=f(N).

5.     This function is not a linear function. This is because the rate of change (slope) varies at every interval. For example, if you take the point (1496,1) representing the 1^st position in the standings and the 1496 laps that person has led through the season, and compare that to the point (195,7) representing the 7^th person in the leaderboard and the 195 laps they led throughout the season. The slope (y2-y1/x2-x1 or change in ranking/change in laps led) of these two points would be: (7-1)/(195-1496) = 6/-1301. If I calculate another rate of change between points (794,3) and (411,8), the slope would be: (8-3)/(411-794) = 5/-383. If you look at the rate of change values for these two intervals they are not the same, which indicates that this function is not linear.

6.     For a function to be a mathematical model, the output values must depend on the input values. This function: R=f (N), ranking as a function of the number of laps led during the season, does not represent a mathematical model. Although theoretically, if one leads more laps, we would tend to think they would be higher in the standings, this is not the case, because the standings are based on points won from racing, not how many laps that driver has led. However, the function: R= f (P), meaning ranking as a function of the number of points a driver had, would be a mathematical model, because the driver’s ranking would directly depend on how many points that driver had received from their races.  

Part B:

1.     Relationships that are not functions are relationships in which one input value gives back multiple output values.

2.     I found a weekly weather forecast in the Washington Post. This forecast has the high and low temperatures listed starting on \ Wednesday October 1, and going sequentially until next Tuesday October 7. Source: http://www.washingtonpost.com/weather/

Wed	Thu	Fri	Sat	Sun	Mon	Tue
--- Mostly cloudy WEDNESDAY Lo:   Hi: 66     74	--- Mostly cloudy THURSDAY Lo:   Hi: 63     76	40% Drizzle FRIDAY Lo:   Hi: 59     76	--- Mostly sunny SATURDAY Lo:   Hi: 53     67	--- Sunny SUNDAY Lo:   Hi: 47     64	20% Chance rain shwr MONDAY Lo:   Hi: 48     70	40% Chance rain shwr TUESDAY Lo:   Hi: 55     70

3.     I am comparing the expected weather condition for the day and the expected high temperature for that day, in other words, the expected weather condition is the input, and the expected high temperature is the output. In function notation, it would be: T(Expected high temperature = f(W) (the expected weather condition).

4.     I know that T=f(W) is not a function because to be a function, every element of one set, in this case, the weather conditions, has to have exactly one corresponding output value- in this case, the expected high temperature. For example, take the input “Mostly Cloudy.” There is more than one corresponding output value(expected high temperature) for the input mostly cloudy, which is 76 degrees Fahrenheit or 74 degrees Fahrenheit. If you plotted these data points on a graph with the weather conditions on the x-axis and the expected high temperatures on the y-axis, this set of points would fail the vertical line test because there would be two dots above the input “Mostly Cloudy.”