# Design A Rollercoaster

## Summary

It took us a long time to come to a consensus on the amount of functions we should use to create a specific roller coaster. We encountered some troubles in coming together, but we managed to meet and form a more concise idea of what we wanted. In our minds, at the time we went towards a more complex approach, favoring complex functions over simpler functions. We had incorporated over 22 functions at the time, as we kept modifying the design. Complexity doesn’t always make better, so we went back to the drawing board to slim down and simplify the coaster.

Some of our group members wanted to keep the complexity, so we settled on keeping a few of them, and removing the extras. We then encountered the lack of polynomial functions, so we decided to remove one sin function, and replace it with a polynomial. We managed to make a roller coaster that fulfilled the requirements and had further functions that added to the overall design. In our last meeting, we were able to make the roller coaster much smoother and finalized the designs.

## General Description

Function 1: This function is a linear model, that starts at y=10, and reaches y=11, at the end. This section lasts for x ϵ R (0, 1.3) seconds. Its slope is increasing in the positive direction. Height will increase with time.

Function 2: This is an exponential function that begins at y=11, and ends at y=16.66. This part lasts from x ϵ R (1.3, 1.8) seconds. This is an exponential growth function since the function is increasing in the positive direction. Height increases, with time

Function 3: The third function is a quadratic function opening down, which is split into half due to time restrictions of {1.885191641x 100 <x< 6.16077}, the quadratic equation starts at y=17 and climbs to y=147 from x ϵ R (1.8, 6.18) seconds. Here as time increases, so does height.

Function 4: This is an even degree polynomial function. It has 2 turning points within its time restriction. This part of the coaster starts at y=147m and ends at 175m. The height increases with time between x ϵ R; (6.2, 11.78) U (21.05, 25.33). The height decreases as time increases between x ϵ R; (11.78, 21.05)

Function 5: Function 5 is another quadratic that’s opening down that starts at x ϵ R (25, 35), starting from a height of y=147 ending at y=60. The restrictions set forth cut the bottom left of the quadratic so it connects to function 4. This part is where the coaster accelerates. Height decreases as time increases as the coaster drops rapidly down.

Function 6: This is an exponentially decaying function. This part of the roller coaster starts at y=57.6m and ends at y=35.8m. The height decreases as time increases, therefore resulting in a negative slop between x ϵ R; (35.5, 40.3)

Function 7: This is a sinusoidal function between x ϵ R (40, 77). The amplitude of the sin function is y=88.4 and is shifted 3.2 to the left. The function started at y=36 and ends at y=78.4

Function 8: This is a rational function and starts at y=221.13 and ends at y= 274.66. It contributes an increase in slope to the ride which means its height increases as time increase between x ϵ R; (73.12, 76.96)

Function 9: This part of the ride is a log function that is exponentially increasing function which initiates at y= 79.6m and ends at y=196.5m. It has an increasing slop at x ϵ R; (60.5, 67.6). Its height increases as time increases.

Function 10: This is a rational function that is increasing, shifted up by 171; it is at x ϵ R; (67.75, 73.5). The function begins at y=197.23 and ends at y=221. As time increases, so does height having a positive incline.

Function 11: This is a tan function but has a large restriction thus makes up very little on the grid, and is from x ϵ R (76.9, 77.23). It begins at y=275.85 and ends at y=279.81. As time increases so does height.

Function 12: This is an even degree polynomial function that is opening downwards, shifted up by 300, at x ϵ R (77.28, 86.12), it begins at y=280 and ends at y=231. As time increases, height decreases.

Function 13: This is a linear function that has a negative slope and is decreasing; the slope starts and ends at x ϵ R (86, 89), it begins at a height of y=230 and ends at y=112. Height is decreasing as time increases.

Function 14: This is another exponential function but is decaying, with a negative slope and is from x ϵ R (89, 100). It starts from a height of y=112 and ends at y=35.9. Height is decreasing, while time increases.

## Equations

1. y= 0.6x+10.25

a. Time Restrictions: {0<x< 1.308598346514}

2. y= 10(x-1) +9

a. Time Restrictions: {1.308598346514<x< 1.885191641x10^0 }

3. y= -7(x- 6.2)^2 +147.00923354

a. Time Restrictions: {1.885191641x 10^0 <x< 6.16077}

4. y= -0.0001(x+1)(x-5)(x-7)(x-15)(x-25)(x-32) + 150

a. Time Restrictions: {6.16077<x< 2.539784706089x10^1}

5. y= -7(x-29.58872)^2 + 300

a. Time Restrictions: {2.539784706089x10^1 <x< 2 3.546654427783x10^1}

6. y= 2^(-x+40) +35

a. Time Restrictions: {3.546654427783x10^1 <x< 4.039081754166x 10^1}

7. y= 38.1sin ((x+3.2))+ 50.3

a. Time Restrictions: {4.039081754166x10^1 <x< 6.047865541721x10^1}

8. y= -210(1 / 8(x-77.4)) +215

a. Time Restrictions: {7.311510189444x10^1<x< 7.696847483577x10^1}

9. y= 200log(x-58)

a. Time Restrictions: {6.047865541721x 10^1<x< 6.769420187481x10^1}

10. y= -300(1/ x-79.1) +171

a. Time Restrictions: {6.769420187481x 10^1<x< 7.311510189444x10^1}

11. y= 50tan ((0.3(x+6.3)) +283.5

a. Time Restrictions: {7.696847483577x 10^1<x< 7.723108691213x10^1}

12. y= -.1(x- 81)^4 +300

a. Time Restrictions: {7.723108691213x 10^1<x< 8.61258438792x10^1}

13. y= -40(x- 91.9)

a. Time Restrictions: {8.61258438792x 10^1<x< 89.14437088}

14. y= 1.5^(-x+99.8) +35

a. Time Restrictions: {89.14437088<x< 100}