Best Roller Coaster By: Abhisek, Bikram, Herman and Kunal

Summary for the Roller Coaster

While we were trying to create our roller coaster we were thinking of making one super complicated with a lot of ups and downs, but reality hit us hard. When we realized that the stuff we wanted to do, wouldn't fit in the time range. We decided to do something a little less complicated, which would fit the time range, and have all the equations we needed. First we decided to put example equations of the seven equations we needed into desmos to get a brief understanding on how it behaved. After we got a basis understanding on how the equations looked, we played around with the different variables in the equation, to see how the graph would change. After, we got the roller coaster forming, we decided to fix up the rough edges, which included overlapping, smoothing the lines and fixing restrictions.

Problems and Solutions for the Roller Coaster

While making the roller coaster, we came across many problems, which included overlaps, smoothing, and fixing restrictions. When we zoomed in on the roller coaster at the start of each equation, we would see an overlapping, that would make the roller coaster have rough edges. When we first seen this, we tried fixing the restriction, but that was not the problem, so we decided to play around with the variables and found out that, that was the problem. We then went ahead to fix each equation, so we wouldn't see any overlaps. After we solved the overlapping issue, we noticed that the rough lines were also solved. Lastly, the last problem we came across was restricting the lines so that the roller coaster flowed properly, and by this we had to play around with the restricting numbers, but we needed to go in the third decimal place to get better results, and get proper connections between each equation.

Descriptions and Equations

Height v.s. Time

General Description: While we were making our roller coaster, we decided to make a scale for the x and y axis that allows us to see the whole grid, within the time and height range, we had to use. After we got that done, we started to plot our equations and making our roller coaster. We also made our roller coaster hit the max point, which is 300 feet and the lowest point we could have which is 12 feet

Equation: Linear Function


- B value is 20, therefore a straight line

-This function is used to make the base of our roller coaster, where the riders will get on

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Equation: Sine Function


- A vertical stretch by a factor of 40 units

- A horizontal compression by a factor of 0.3 units

- A horizontal shift by 4.42 units to the right

- A vertical shift up 80 units

- This equation was used to make a drop and rise for our second drop

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Equation: Polynomial Fuction


- A vertical compression of 0.00005 units

- A vertical shift 94.5 units up

-This equation is used to connect our first drop all the way up to our second drop, also we used it as a fifth degree because we wanted a lot of ups and downs that were a lot less extreme

- We used a factor with a power of 3 to have the roller coaster approach the x-intercept and go back down again after touching it

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Equation: Logarithmic Function


- A vertical shift 40 units up

- A horizontal shit 82.831 units to the right

- A base close to 0, because it would rapidly curve the line towards infinity in the x

- This equation was to used to create a downwards slope for the roller coaster, and we used a number between 0 and 1 to do so

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Equation: Rational Function


- A vertical stretch of 100

- A horizontal shift to the right by 84.415 units

- A vertical shift 128.455 units up

- This equation creates a drop for our roller coaster, which is the last drop better it ends

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Equation: Quadratic Function


- A reflection about the x-axis

- A vertical stretch by a factor of 2.3

- A horizontal shift 28.5 units to the right

- A vertical translation of 300 units up

-This equation was used to make the arc of the roller coaster, where it would climb and then drop from

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Equation: Exponential Function


- A vertical stretch by a factor 1.27055

- A vertical translation 18 units up

- This equation was used to create our first rise for the roller coaster

- Exponential growth

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Solving for time at 12 feet:

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Solving for times at 250 feet:

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Average rate of change between 10 to 15 seconds:

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Average rate of change between 50 to 60 seconds:

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Instantaneous rate of change at 35 seconds:

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Final Outcome

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