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Horizontal and Vertical Asymptotes

Guidelines that graphs approach based on zeros and degrees in rational functions.

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A Shot in the Arm

Credit: Tanner Lange/U.S. Navy
Source: http://commons.wikimedia.org/wiki/File:US_Navy_060407-N-4015L-003_Hospital_Corpsman_2nd_Class_Lawrence_Osborne_administers_a_tetanus_shot_to_A_Sailor_during_an_immunization_stand_down_aboard_the_Nimitz-class_aircraft_carrier_USS_George_Washington_%28CVN_73%29.jpg
License: CC BY-NC 3.0

Lockjaw is a terrible way to die. As your body reacts to the toxins coursing through your blood, your muscles lock up. Violent spasms wrack your body. You can’t eat or drink. If you survive, you’ll probably suffer from permanent nerve damage. And, 100 years ago in the United States, you probably weren’t going to survive a case of lockjaw. Now, the disease is very rare. How did this happen?

Nearly Eliminated

Credit: CDC
Source: http://www.cdc.gov/tetanus/surveillance.html
License: CC BY-NC 3.0

Lockjaw, also known as tetanus, used to be a common disease in the United States. Today, there are only a couple of deaths from the disease a year. If you look at the graph above of tetanus cases and deaths in the U.S. (per 1 million people) from 1947 to 2008, you can see that they both follow the pattern of rational functions with an asymptote at \begin{align*}y=0\end{align*}. Tetanus cases decreased very rapidly at first, but then its decline slowed down. There are still a few deaths from tetanus every year, but always fewer than 10; most years there are fewer than 5.

Credit: Andreanna Moya
Source: http://www.flickr.com/photos/andreanna/2975644137/
License: CC BY-NC 3.0

What changed? The tetanus vaccine became common in the U.S. as a result of successful public health campaigns. Clostridium tetani, the bacteria that causes tetanus, is everywhere in the environment. It can thrive inside wounds, especially puncture wounds. You can get it from being poked with a rusty nail or even by a thorn in the garden. If you’ve been vaccinated for tetanus and your shots are up to date, your body is prepared to attack the infection. If the infection runs wild, however, it fills your body with a poison that destroys your nerve cells and causes the life-threatening symptoms. Though most Americans will never need to think about tetanus, people in the developing world today still have reason to fear lockjaw and its terrible effects.

See for yourself: http://www.youtube.com/watch?v=jPUQakhyPDA

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Watch the video below to learn about Richard, a six-year-old boy from Uganda, and his tragic battle with tetanus. At the next two links, learn about vaccination efforts in the Ivory Coast and Kenya.




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