# Kage Bunshin No Rollercoaster

## Creation Process of the Rollercoaster

Creating the rollercoaster was no easy task, as it required time and patience to complete. We had initially started by compiling a list of all the parent functions and using that knowledge we had sketched out an idea of how we imagined our final graph to turn out. We had to take many factors into consideration, such as the maximum and minimum points, the degree for some functions for example, polynomial functions, as well as the time limit for the entire graph. Next, we tried to apply transformations to the equations of the functions and restrictions in the domain and range of their equations. In order for us to correctly visualize how our graph was turning out we started using the graphing software called Desmos. In the end, we had finally come up with all of our equations, with the restrictions and it had seemed to connect perfectly, but after closer inspection we had realized that some of the equations were not actually attached to each other, or they were over lapping each other. With this in mind, we went back to Desmos and started tweaking the equations in order for them to connect properly.

## Written Report

In our graph, time is an independent variable and height is a dependent variable. Throughout the graph, time is always increasing; however, height is the variable that keeps fluctuating. The graph starts at (0, 10) with both time and height increasing at the same time, this continues until 20s, where time is still increasing but the height of the graph starts to decrease. At 31s the height of the graph starts to increase, until at 34s it drops and pulls up again at 41.5s and once again drops, until it rises back at 43s. Afterwards, the height increases exceptionally faster than time, between 43s and 45s.

The height of the graph is increasing but almost seems to be at constant, between 45s to 60s. At 67.5s, our graph reaches its maximum height of 300ft. After reaching the maximum point, the height falls drastically to 10ft, while time is still increasing, until it reaches 100s.

Equations and their restrictions

y = 2^x+9{0≤x≤6.06}

y = -(x-20) ^2+270{6.06≤x≤30}

y = -35sin(x-30) +170{30≤x≤43.2529963582}

y = -1/x-45+147.24{43.2529963582≤x≤45}{147.81≤y≤250}

y = log(x=44.990268) +264.148{44.6≤x≤59.9825}

y = 4.34x+5{59.9825≤x≤62.8593}

y = -(x-67.57)^2 +300{62.8593≤x≤69}

y = -(x-68) ^3 +299{68.789≤x≤72.938804}{184.735<y}

y = 90 ^(x-74) +10{72.8≤x≤100}{184.735>y}

## Rollercoaster Graph Design

Y- Axis : Height (feet)

X- Axis : Time (seconds)

Max Height : 300 feet

Min Height : 10 feet