# Coaster DysFUNCTION

### By: Krutarth Dave, Rut Parikh, Karan Sharma and Grace Bang

**NOTE:** Keep in mind that the graph on Desmos does not abide with the Legend above!

## How We Created This Rollercoaster

We began by creating the base of the roller coaster, where it would begin and where it would end. We figured that most roller coasters do not start on the ground, so we ascended it by 10 feet. In reality, this would be where individuals take the flight of stairs to the boarding station of the roller coaster; this was the first point on our height vs. time graph of our roller coaster (10 feet at 0 seconds). Next, we decided to end our roller coaster at a height of 12 feet or higher; this was done to keep a safe distance from a height of 10ft. (as the graph was required not to go under 10ft.). Deciding these endpoints of the coaster, made it easier to create the shape of the coaster. Keeping this in mind, we began to roughly design our roller coaster by sketching it out on paper attempting to incorporate all seven of the required equations (see the rough sketches below; the sketch with the check mark is what became the final copy). We knew that we needed to create functions that matched the rough sketch and fit together effortlessly. In order to do this we set the restrictions on the equations in such a way that the beginning of the restriction of the next equation, would be the same value as the end of the restriction of the previous equation. Next, we used trial and error and our knowledge of advanced functions, and adjusted the appropriate transformation variables of each equation so that the coaster had a continuous flow.

## Difficulties We Faced

The first hurdle we had slight difficulty in overcoming was creating the rough drafts of the graphs. We were confused at first, of how to incorporate all seven equations and determining an appropriate order for the equations so that our roller coaster would “flow”. After we made the rough draft of the roller coaster, the next difficulty we faced was figuring out how we would place it onto Desmos. At first, coming up with each equation for Desmos seemed like a hard task but after the first two equations, we had a good idea of what we were doing. We realized that we just needed to review our notes a little to remember some things that we might have forgotten (such as which variables control which transformation). The next difficulty we faced was the major one. When we zoomed into our graph on Desmos, even a tiny bit, we could see gaps in our roller coaster. In order to fix this, we zoomed into the graph on Desmos to its full extent. Next, we transformed all the equations (in left-to-right order) by minuscule amounts in order to rid the graph of any gaps. This why the equations (below) contain lengthy strings of decimal values.

## Equations

## Calculations!

**NOTE:**The long strings of decimal values (seen in the equations above) were rounded to the nearest tenth for the ease of conducting the following calculations:

## Time(s) at which the graph is at 250ft.:

## Equation 1 (Parabola) This equation represents a parabolic function, and we solved for the time at which the equation was at a height of 250 ft. The function reaches a height of 250 ft. at 7.041 seconds. | ## Equation 2 (Sinusoidal) This equation represents a sinusoidal function, and we solved for the times at which the equation was at a height of 250 ft. The function reaches a height of 250 ft. at 17.27s and 27.74s. | ## Equation 4 (Exponential) This equation represents a exponential function, and we solved the time for which the equation was at 250 ft. Since there is only once possible value for height, restrictions did not need to be considered. The function reaches a height of 250 ft. at 37.64s. |

## Equation 1 (Parabola)

## Equation 2 (Sinusoidal)

## Summary of time(s) at height of 250 ft.:

## Time(s) at which the graph is at 12ft.

## Equation 1 (Parabola) This equation represents a parabolic function, and we solved for the time at which the equation was at a height of 12 ft. The function reaches a height of 12 ft. at 0.041 seconds. |

## Summary of time(s) at height of 12 ft.:

## Average Rate of Change / Instantaneous Rate of Change

For the 10 - 15 seconds and 50 - 60 seconds intervals, the average rate was calculated by:

- First, checking the restrictions set on all graphs and determining which equations are part of this interval.

- Then substituting the appropriate time values for "x" in the appropriate equations (determined in the first step) to solve for height

For the instantaneous rate, 35 was substituted for "x" in the appropriate equation and used to solve for height. A new value slightly higher than 35 (35.001) was substituted again into the same equation to find a "new-height". The slope between these two points was calculated to find the instantaneous rate.

## Average Rate of Change Between 10s-15s Taking the restrictions that are set on the equations into consideration, we came to the conclusion that the 10-15s interval was entirely in Equation 1, which is the parabola. | ## Average Rate of Change Between 50s-60s By taking the restrictions that are set on the equations into consideration, we came to the conclusion that the 50-60s interval was between Equation 5, which is the logarithmic function, and during Equation 7, which is the linear function. | ## Instantaneous Rate of Change at 35s Taking the restrictions that are set on the equations into consideration, we came to the conclusion that at 35s, the equation was entirely in Equation 4, which is the exponential function. |

## Average Rate of Change Between 10s-15s

**Equation 1**, which is the parabola.

## Average Rate of Change Between 50s-60s

**Equation 5**, which is the logarithmic function, and during

**Equation 7**, which is the linear function.

## General Description of Coaster DysFUNCTION's height vs. time

The Coaster-DysFUNCTION starts at a height of 10’ and gains height at a rapid pace until it reaches the height of 300’ in a matter of 12.04 seconds. During this climb, the roller coaster initially has a very high rate of change which gradually decreases as more height is gained. At the instantaneous moment at 12.04 seconds, the coaster is at 300’ and is neither gaining, nor losing, height – meaning that the rate is equal to zero (horizontal tangent). After the 12.04 second mark, the coaster begins to lose altitude until it reaches the height of 200’. During this decline, the coaster accelerates downwards (increasing rate) until about 19.80 seconds; after which the rate begins to decrease. At the instantaneous moment at 22.51 seconds, the rate is zero and the coaster stays at a height of 200’. Next, the graph begins to rise again and continues to do so for another 8.49 seconds (31 total seconds), until it reaches a height of 275’. During this time, the rate of the coaster is initially increasing until about 27.74 seconds after which it begins to decrease again. At the 31 seconds mark, the rate of change is zero. Afterward, the coaster begins to lose height until it reaches a height of 53’; when this height is achieved, 53 seconds have passed from the moment that the coaster began. While the coaster is losing height in this interval, the rate increases until approximately 40.45 seconds; the rate begins to decrease after that point as the coaster approaches 53’. Progressing forward, the coaster begins to rise once again until it reaches a height of 200’; when this height is reached, 67.88 total seconds have passed. During this final rise of the coaster, its rate is initially increasing (between 53 seconds and 59 seconds); then it continues to rise at a constant rate (of 30’/s) (from 59 seconds to 61 seconds). Between the 61 second mark and 67.88 second mark, the coaster’s rate is decreasing as it reaches its final “high-point” (local maximum) – at which point the rate is zero (height: 200’). Finally, during the concluding 32.12 seconds of the Coaster-DysFUNCTION ride, it loses altitude from 200’ and comes to an end at a height of 12’ (final time: 100 seconds). Throughout this decline, initially the coaster’s rate is increasing, until approximately 74.07 seconds, then it begins to decrease. When the coaster reaches the height of 123’ (at 80 seconds), its rate of change is zero. After 80 seconds, the rate begins to increase again, until about 89 seconds; then it begins to decrease again for the final phase of the coaster as it approaches and stops at a height of 12’ at a time of 100 seconds.