Highway to Hell
By: Amrita Kumar & Mihir Kapadia
The Starting Stages
Before even starting to create our rollercoaster, we knew we would have difficulties with the rational, exponential and logarithm functions as they each have restrictions. Understanding this, our knowledge from our Advanced Functions class, and previous math classes, guided us through the process of creating our rollercoaster on Desmos.
We first tried to use 'base functions' with minimal transformations to make the rollercoaster simple and smooth, however that did not work for all of the functions because they would either go past the maximum height or they wouldn’t reach it. We wanted to touch the maximum height at least one time in the duration of the ride, so in order to determine the formulas for all of our functions, we generally took the end point of one function and substituted it into the next function to see where the point of intersection would be. In addition, we modified the transformations accordingly so that the rollercoaster would not go past the maximum or minimum height.
A difficulty we came across was in terms of how our drafted rollercoaster physically differed from our final rollercoaster. We found that we could not always find the exact intersection point between two functions, unless we wanted to go into many decimal places, and that our final rollercoaster was not as 'smooth' as it would realistically be.
Functions & General Descriptions
Linear h(t) = 10 -Slope: 10 D: 0 is less than or equal to X which is less than or equal to 1 | Exponential h(t) = 2^t + 8 -VT of 8 units up D: 1 is less than or equal to X which is less than or equal to 7.91795 | Quadratic h(t) = -t^2 + 30t + 75 OR h(t) = - (x - 15)^2 + 300 -VR in x-axis -VT of 300 units up -HT of 15 units right -Vertex: (15,300) D: 7.91795 is less than or equal to X which is less than or equal to 29.6965 |
Exponential
-VT of 8 units up
D: 1 is less than or equal to X which is less than or equal to 7.91795
Polynomial h(t) = (x - 30)^4 +84 -VT of 84 units up -HT of 30 units right -Vertex: (30, 84) D: 29.6965 is less than or equal to X which is less than or equal to 33.495805 | Sinusodialh(t) = 70 sin[0.1 (t + 30)] + 228.7 -a= 70 -k= 2Pi/0.1 =20Pi -phase shift of 30 units left -Equation of Axis (c) at 228.7 D: 33.495805 is less than or equal to X which is less than or equal to 79.8 | Rationalh(t) = 158t^2 + 300 / t^2 + 1 -Y-Intercept: (0, 300) -HA: x = 158.7 -No VA D: 79.7036642 is less than or equal to X which is less than or equal to 89.6 |
Polynomial
-VT of 84 units up
-HT of 30 units right
-Vertex: (30, 84)
D: 29.6965 is less than or equal to X which is less than or equal to 33.495805
Sinusodial
h(t) = 70 sin[0.1 (t + 30)] + 228.7
-a= 70
-k= 2Pi/0.1
=20Pi
-phase shift of 30 units left
-Equation of Axis (c) at 228.7
D: 33.495805 is less than or equal to X which is less than or equal to 79.8
Logarithm h(t) = -220 log(t - 87) + 250 -VR in x-axis -VS of 220 -VT of 250 units up -HT of 87 units right -VA at x = 87 D: 89.5996 is less than or equal to X which is less than or equal to 100 R: Y is greater than or equal to 10 | Linear h(t) = 10 -Slope: 10 D: 99.3 is less than or equal to X which is less than or equal to 100 | Final Look h(t) = 10 h(t) = 2^t + 8 h(t) = -t^2 + 30t + 75 OR h(t) = - (x - 15)^2 + 300 h(t) = 70 sin(0.1t + 3) + 228.7 h(t) = 158t^2 + 300 / t^2 + 1 h(t) = (x - 30)^4 +84 h(t) = -220 log(t - 87) + 250 h(t) = 10 |
Logarithm
-VR in x-axis
-VS of 220
-VT of 250 units up
-HT of 87 units right
-VA at x = 87
D: 89.5996 is less than or equal to X which is less than or equal to 100
R: Y is greater than or equal to 10