Wheres the Function?

I found online President Obama’s approval ratings listed every day spanning the entire length of his presidency thus far. The website is :

http://www.realclearpolitics.com/epolls/other/obama_bush_first_term_job_approval.html . This is a function because for every input, the day, there is one matching output, the approval rating. It therefore passes the vertical line test. In this case, the President’s approval rating is the dependent variable. It depends on the independent, which is time (in days). The approval ratings depends on the time because over time the President makes certain decisions that can sway his ratings. One month he may have a spike in ratings due to a successful initiative and another month he may have a drop due to a blunder, like his recent MediCare site. The function in function notation would be:

 Approval Rating=f(Time)    à time in days

This function is NOT linear, because the values of the average rate of change are not identical. If the function were linear, the slope would be consistent and create a straight line when graphed. Approval ratings are variable that depend on random, independent occurrences rather than predictable, stable variables. If the function were linear, the “changes in rating” would all be the same, indicating a steady linear relationship.

            As stated earlier, this function is a mathematical model. Although some could consider it isn’t, I argue that it is. My reason being is that in this case, time in days is not just a physical date, but it is an indication of choices being made by the President. On January 17, 2014 the President has extremely poor approval ratings following his Medicare blunder. But his ratings may have been higher following healthcare reform initiatives.

            As for relationships that are not functions, I struggled to find an example but might have gotten one. When reading about professional sport’s teams uniforms, I caught something. Functions CANNOT have the same x-values. Sure, if we are looking at New York Yankees Player, Shirt Color being (x, y), this is a function. But if we change it around to Shirt Color, Player , this is no longer a function. A FUNCTION MAY NOT HAVE TWO Y-VALUES ASSIGNED TO THE SAME X-VALUE.

The article I was looking at was:

http://sports.yahoo.com/news/top-25-uniforms-worn-professional-sports-teams-years-185100394--spt.html

The relationship I was looking at was between Professional Athletes and shirt colors. 

Where's the Function?

1.http://www.calculatedriskblog.com/2013/10/hotels-occupancy-rate-tracking-pre.html

 This article is describing the rates of occupancy of hotel rooms during different times of the year. I thought this was an interesting take because when I think of occupancy rates I think it would more depend on the number of hotel rooms available. I also liked that this graph showed the different rates compared to other years-this way the graph was not too thrown off from the recent recession.  The graph represents a function because the one output is directly affected by one input . The changing weeks of the year make a difference on the occupancy rates for hotels. Therefor, the graph is also a mathematical model because the output is determined by the input.

2. http://www.cbsnews.com/news/majority-of-americans-now-support-legal-pot-poll-says/

This article describes how a majority of Americans now favor the legalization of pot. However, the relationship between being an American and favoring the legalization of pot is not a function because there are multiple outputs for the same input. Just because you are an American does not mean you believe pot should be legalized. 

Blog 2

SAT mean scores of college-bound seniors, by race/ethnicity: Selected years, 1986-87 through 2010–11
Year	1986–87	1990–91	1996–97	1999–2000	2000–01	2001–02	2002–03	2003–04	2004–05	2005–06	2006–07	2007–08	2008–09	2009–10	2010–11
SAT- Critical Reading	507	499	505	505	506	504	507	508	508	503	502	502	501	501	497
SAT- Mathematics	501	500	511	514	514	516	519	518	520	518	515	515	515	516	514
SAT- Writing	†	†	†	†	†	†	†	†	†	497	494	494	493	492	489

https://nces.ed.gov/fastfacts/display.asp?id=171

These statistics come from the National Center for Education Statistics. The table above displays the average SAT scores for the corresponding years. The table above represents a function because for every input, of years, there is an output, of scores. However, it is not a linear function as the rate of change is not constant.  It fluctuates yearly either decreasing or increasing meaning that the entire table, should it be graphed, would not have a constant negative or positive slope. This is not a mathematical model because the SAT scores do not depend on the year. Just because the academic year is 2003-2004 does not affect the scores, whereas other factors would.

A function is determined if each input has exactly one output. Therefore in order for a relationship to not be deemed a function there must be more than one output for each corresponding input. An example would be of,  he tide levels on the Potomac river. If the input is one day, say Tuesday, and there are four different readings for Tuesday, then this is not considered a function. If there are four outputs, of different tidal levels, during the one input period of Tuesday, then the information presented is not a function. For Tuesday alone the tide was -0.05 feet and 2.53 feet.

Washington, Potomac River, D.C.
Tuesday	01/21/14	5:16PM	Sunset
Tuesday	01/21/14	6:04PM	-0.05 feet	Low Tide
Tuesday	01/21/14	10:44PM	Moonrise
Tuesday	01/21/14	11:24PM	2.53 feet	High Tide
Wednesday	01/22/14	6:14AM	-0.18 feet	Low Tide
Wednesday	01/22/14	7:21AM	Sunrise
Wednesday	01/22/14	10:25AM	Moonset
Wednesday	01/22/14	11:43AM	2.66 feet	High Tide
Wednesday	01/22/14	5:18PM	Sunset
Wednesday	01/22/14	6:49PM	-0.02 feet	Low Tide
Wednesday	01/22/14	11:44PM	Moonrise
Thursday	01/23/14	6:56AM	-0.17 feet	Low Tide
Thursday	01/23/14	7:21AM	Sunrise
Thursday	01/23/14	10:59AM	Moonset
Thursday	01/23/14	12:10PM	2.48 feet	High Tide
Thursday	01/23/14	12:30PM	2.7 feet	High Tide
Thursday	01/23/14	5:19PM	Sunset
Thursday	01/23/14	7:42PM	0.02 feet	Low Tide

http://www.tides.info/?command=view&location=Washington%2C%20Potomac%20River%2C%20D.C.