# Design a Rollercoaster!

## The VINICoaster!!

Purple: Exponential Function

Black: Sinusoidal Funtion

Red: Quartic Funtion (polynomial function)

Green: Reciprocal Function

Orange: Logarithmic Function

Red: Linear Function

The above picture is a screenshot of the roller coaster, designed using desmos, which is composed of 7 different functions of the many that were covered in the Advanced Functions course. However this roller coaster wasn't created on the first attempt, the picture below is the rough graph before we made a few changes to the final roller coaster.

## Summary of Design

Research was conducted to find examples of 2-D roller coasters to inspire our own roller coaster design. We set out a plan for a general design for our roller coaster. We then implemented the 7 required functions into our roller coaster design. Each function was created in such a way so that there is a point of intersection between each consecutive function, and each function smoothly transitions to the next. In other words, the IROC at each intersecting point between functions is similar if not the same. We restricted the domains of each function to start and end at the points of intersection between the functions (restriction for first and last function at x=0 and x=100). We were able to use algebra and families of functions to make functions intersect at desired points, however we had trouble with trying to make sure the functions intersected such that the IROC was similar at the points where the 2 functions connected as we had to make sure that the points where 2 functions intersected had similar or same IROC. The biggest difficulty we had was experimenting with the polynomial function, it took many attempts, trying different values until the polynomial function looked ideal for a roller coaster. Overall we were satisfied with the final design of our roller coaster.

## General description of functions of height vs. time

Height vs Time functions describe the vertical position of an object at specific times. This means that they can also be used to describe the change in vertical position (vertical displacement) over a time interval, which is known as the vertical velocity. A Height vs Time function is a function of time (time is the independent variable), this means that there can never be two y values that share the same x value because that would mean an object is at 2 different positions at the same time which is impossible. However it is possible to have multiple x values that have the same y value, because an object is allowed to come to the same position multiple times. The slope of a Height vs Time function represents the vertical velocity. Negative slope represents going down, positive slope represents moving up, a decreasing slope represents downward acceleration and an increasing slope represents the upward acceleration. Overall a Height vs Time function can describe more than just the position of an object.

## a(x)=2^(x-7)+10 {0<=x<=14.17}

• a(x) is an exponential function that was chosen as our first function because we wanted to start with a function that had increasing slope.
• The reason we wanted the increased slope is because we wanted our roller coaster to accelerate vertically in the beginning of the ride to make it a scarier ride.

## b(x)=-(2.5(x-19))^2+300 {14.17<=x<=23.9}

• b(x) is a negative parabola used to create the peak for the first hill of our ViniCoaster.
• The vertex of this parabola represents the highest point of the ViniCoaster.
• The parabola also represents that the roller coaster stops accelerating vertically as it reaches the peak and actually begins to slow down, until it goes back down again.

## c(x)=-50(cos 30(x-2.9))+150 {23.9<=x<=41.9}

• c(x) is a cosine function used to make our second drop not as high and as long.
• We used a sinusoidal function because we wanted consecutive drops that would have the same minimum height.

## d(x)=-(1/100)((x-41.9)(x-50)(x-60)(x-70))+150 {41.9<=x<=68.902}

• We included this quartic fuction to allow our roller coaster to have progressively larger hills.
The 4th peak is larger than the 3rd.
The 3rd peak is larger than the 2nd.

## f(x)= [1/((0.005)(x-66.9))]+100 {68.902<=x<=79.9}

• This function is used as a smooth transition into the rest of the roller coaster after the previous drop.
• This is better than having a sharp end to the previous drop.
• We used a reciprocal function because it is one of the few functions that have a long and slight horizontal curve

## g(x)=-50.2log (x-75)+150 {79.9<=x<=90}

• This log function continues to ease the ride into final phase of the roller coaster, which was the final drop.
• This part of the roller coaster acts as a gentle transition.
• We used a log function because it is one of the few functions that have a long and slight horizontal curve

## h(x)= -8.1x+820 {90<=x<=100}

We used a linear function to end the ride and bring the ride back to the minimum height .