# data[].attributes.body
These are the equations I used to graph my piece-wise function!
(The cut of numbers in the brackets are my domain restrictions!)

# data[].attributes.body
Exponential decay is a type of exponential function that goes from a higher value (x,y) to lower value. In a graph, it looks like a usually shown exponential graph -exponential growth- (open toward the left), but opposite (opening to the right). Exponential decay shows the decrease in the subject over time, as exponential growth shows the growth over time. My graphs show the exponential decay of caffeine (y) over time (x).

# data[].attributes.body
"Caffeine is a central nervous system stimulant. It can temporarily make you feel more awake and energetic, but it can also give you the jitters. Withdrawal or overdose can cause a range of problems."

# data[].attributes.body
Caffeine is a central nervous system stimulant that temporarily makes you feel more awake and energetic. My research shows that the effects of caffeine definitely correlate with sleeping patterns. Caffeine is even still in the system at such high numbers while sleeping, that could affect the sleep. I recommend that to allow sufficient and efficient sleeping, you should stop drinking caffeine at least 5 hours before you plan on sleeping to allow time for the caffeine to exit the system. You could also track your own caffeine consumption! Doing this and using the equation y = c (0.87)^x could help you determine what time would be best to help you sleep better. In this equation, c is the amount of caffeine consumed, 0.87 (or 87%) is the amount of caffeine left in your body after an hour, and x is the time, or how many hours. Y  will be how much caffeine is left in your body after x hours. Drink well, but not too much to have extreme negative effects on your body and your sleeping!

# data[].attributes.body
3.

# data[].attributes.body
2.

# data[].attributes.body
1.

# data[].attributes.body
These images are of the same graph, closing in on the detail so it is easier to see. The highlighted areas are the sleep-times over this 48 hour period of data collection.

# data[].attributes.body
Next, the graph below shows a piece-wise function of the caffeine consumption of a family member over 48 hours.
(y- caffeine in mg; x- time)

# data[].attributes.body
Below, you will see a chart with the data of caffeine consumption of a family member over a 48 hour period.

# data[].attributes.body
In the next graph, the point shows when on the graph the initial caffeine be reduced to an amount that should no longer have any stimulant effect on the body (when the initial is still 80 mg of caffeine). At about 6 and a half hours, the caffeine level in my body will be at 32mg (6.5,32), and depending on the person, lower caffeine levels after 6.5 hours should no longer have any stimulant effect on the body.

# data[].attributes.body
This equation

{ y = 80 (0.87)^x }

can be used to model how much caffeine is left in your system.
This is a graph of the equation above

# data[].attributes.body
"How Long Does Caffeine Stay In Your System?" Mental Health Daily. N.p., 30 Oct. 2015. Web. 21 Jan. 2017.

"17 Effects of Caffeine on the Body." Healthline. N.p., 22 Oct. 2014. Web. 21 Jan. 2017.

# data[].attributes.body
1. What type of function can you use to represent the amount of caffeine in your body?  Explain.

1. You can use an exponential function to represent the amount of caffeine in your body because it shows growth or decay of the variable over time.

2. Why would a linear function not make sense to represent the amount of caffeine in your system?

1. A linear function would not make sense to represent the amount of caffeine in your system because a linear functions are straight lines that only increase or decrease, or showing no fluctuation in data. In caffeine consumption, the caffeine level in your body starts high, and won’t decrease in a straight line on a graph. You could also consume more caffeine and the level will rise and lower again.

3. When will the caffeine completely leave your body?

1. Depending on the individual, it would take "less than 8.25 hours, whereas others may take up to ~2.18 days" for the caffeine to completely leave the body. These mostly depend on the person and their characteristics, such as age, metabolic rate, and body weight and height.

4. When will you have half of the caffeine left?  How long does it take for the first half to disappear?  How long for the second half?  Show this algebraically.

1. When having 80 mg of caffeine in my system I will have about half of the caffeine left in my system (39.87 mg) after about 5 hours, written as y = 80 (0.87)^x-5. The second half is almost out of my system at 32 hours (at 0.92mg caffeine; 0.1mg at 48 hours;  0.00000003mg at 155 hours).