# Nephron

## Development

Originally, we wanted to make a graph, and ultimately, rollercoaster, that started at 300 feet. Planning for this graph, we made our first rough draft:
As you can see, this graph starts at the top (300 feet). After realizing the potential hazards and the lack of funding this idea had the possibility to achieve, we intended to go to a safer route, our final graph. Similar to the original, but this time starting at 10 feet instead of 300 feet, our new graph was akin to the likes of a traditional rollercoaster. As with the original idea, we made a second rough draft to match out new rollercoaster idea:
The marks (dashes) made on the rough draft are for dividing potential functions. At the end of the process, the rough draft helped a lot.

## Summary

Designing the Nephron was a challenge. Originally, many designs were created but the most interesting was the most unique. As a group, we agreed to create the roller coaster starting at a height of 300 feet (the first of its kind). However, due to cost/potential lack of funding a rollercoaster that starts that high, we were forced to flip our original graph and made the rollercoaster start at ten feet. The basic shape and where to put all the functions were already known thanks to the rough sketch. The first challenge faced was to design the rollercoaster on Desmos. The rollercoaster was built from left to right. Each functions domain was carefully restricted to make sure there were no rough edges. Sometimes, it was hard to make the functions intersect and careful calculating, planning, and precision were needed to make it work. Another challenge was to make the roller coaster run smoothly. Some functions would intersect at sharp angles and quadratic equations were placed in between them to offer a smoother ride. The final challenge faced was to make sure the functions intersected in the first place. The a, k, c and d values were all changed in order to make them intersect perfectly. That proved to be the most difficult challenge when graphing. Furthermore, making the graph again was a challenge. Since starting at 300 feet, originally, presenting funding and resource problems, we decided to make the graph start at 10 feet. This took more time and needed additional calculations, but was accomplished.

## Height vs Time

For each function, there is a relationship between height and time. Time is the x axis (x variable) while height is the y axis (y variable).

To give the whole graph on one scale, we changed the settings to only include the graph inside the range and domain (within 100 seconds for time and within 300 feet for height). That means we only showed the grid until the 300 feet and 100 second mark to make things easier.

Additionally, the importance, purpose, and general descriptions of the functions are described below:

## Linear Function

Please note that the domain means RESTRICTIONS on time (x)

Equation, Domain

Range: {37.3<y<120}

The linear function represents the roller coasters attempt to come back to its starting point It gradually decreases, by making sure that it is not too steep, we ensure that the passengers can return back to the starting point safely without any doubts.

-a value is 1

-b value is 120, so it starts 120 feet above the ground

Equation, Domain

-No compressions or stretchs (factor of 1)

-Vertical translation 10.999 feet up

-Horizontal translation 10.21 units to the right

Range: {11<y<254}

The quadratic function represents the climb that passengers of the roller coaster experience during the first portion of the ride. They gradually rise higher and higher at a quicker rate until they come roaring back down.

## Polynomial Function

Equation, Domain

Range: {257<y<300}

The polynomial function represents itself as an addition to the quadratic function. Since it is a quartic function, the closed portion of the parabola is larger, allowing the passengers to get a view of everything around them, and without hesitation, causing them to fall right back down.

## Rational Function

Equation, Domain

-A vertical translation 301 feet up

This equation serves as the link between the most crucial moments of the rollercoaster, the rise and, most importantly, the fall. This equation gives the riders a moment to feel peace and quiet at the absolute top of the rollercoaster as the ride moves forward a the top for a brief moment, before plummeting down from 300 feet.

## Sinusoidal Function

Equation, Domain

-No vertical stretch (factor of 1)

-Horizontal stretch by a factor of 3.3 (factor of 1/k)

-Reflection in the x axis

-Vertical translation 11 feet up

-Horizontal translation 1 unit right

Range: {oo<y<12}

In the very beginning of the ride, this sinusoidal function serves the purpose of moving the passengers to, arguably, the most important part of the ride, the hill to the top. This equation also has a very slight change in height as there is, purposely, a slight bump to give the riders a feel for the rollercoaster journey before the most exciting parts.

The second sinusoidal function is connected to the first. Both are very similar as the provide the same purpose, to connect the track to the hill portion of the ride. The second sinusoidal function is:

## Exponential Function(s)

Range: {y<36.942356388}

-Exponential decay (since base is under 1)

-Vertical translation 10 feet up

-Horizontal translation 97 units right

This was the last equation in the rollercoaster right/graph as it provided the bride to the beginning of the rollercoaster. Since rollercoasters, traditionally, are connected from on end to the other, there needed to be an equation to connect the two sides. Not only does the exponential decay function serve a purpose for slowing down the ride, it also, as mentioned before, served the necessary purpose of completing the rollercoaster, ending it at 10 feet where the passengers can leave after an exciting ride.

Exponential Function #2

-Exponential Growth

-Exponent base 2

-Reflection in the y axis

-Vertical translation up by 300 feet

-Horizontal translation to the right by 40 units

Starting at 300 feet and lasting for approximately 12 seconds, this exponential function serves the purpose for the drop in the rollercoaster. Arguably the best part of the ride, the exponential function was necessary for the drop because it provided a steepness to make it more exciting.

Exponential Function # 3

This function is where the passengers of the rollercoaster initially board. The domain of the function is relatively small because the equation only needs to board the passengers onto the ride. The remaining rollercoaster ride is connected to this initial, and first equation:

## Logarithmic Function

Equation, Domain

-Vertical translation 205 feet up

-Horizontal translation 45.9 units to the right

-We chose a base under one because we needed the right amount of curve to connect the log equation to the exponential.

Range: {161.76<y<219.92}

Serving a crucial purpose as it follows the drop, the log function needs to safely lower the velocity of the rollercoaster to make it safe for the passengers and make it rideable. We made the log equation, keeping in mind the necessity for a slightly longer curve, to slow the ride down and to provide a duller angle, to prevent injuries. If the log function was made to have a sharper curve, a higher angle from the fall would mean that the ride would potentially cause harm, therefore receiving no support and funding. Immediately following the drop, the log function serves a very important purpose.

## Calculate Average Rate of Change

*Note: The AROC is in Ft/Second because of the height vs time relationship (height is in feet and time is in seconds)