THE BRIGNULLATOR
Final Summative: RollerCoaster Model
MHF 4U0  Brian, Nicholas & Srushti
Summary:
In the creation of our rollercoaster model, we incorporated thirteen different equations, which consisted of either a linear, quadratic, polynomial, rational, sinusoidal, exponential, and logarithmic function. The initial objective was to keep our rollercoaster fun, different, and most of all realistic. Our first step in creating our rollercoaster was placing the given maximum and minimum height, and also incorporating it in the given time of 100 seconds. Next, we added in the functions into the graphing calculator one by one. As we add each function, we discuss the appropriate transformations and restrictions where each function would be suitably fit. It was also important to make sure the requirements were met, such as have a function at the maximum height of 300 feet and a minimum height of 10 feet. As we arranged each function, we made sure that it was realistic in terms of utility and had no mathematical errors. In order to make the rollercoaster practical, we concluded that the rollercoaster would start at the minimum height of 10 feet then incline to the maximum height of 300 feet and finally end off at the minimum height. Throughout our process, we had one main difficultly which was, determining where each function would be placed, this was because each function had its own uniqueness which would benefit our design in numerous ways.
Description of Functions:
Exponential Function h=2^t+9
 Quadratic Function h= 4 (t12)^2+300
 Logarithmic Function h= 50log (t15.5)+200

Exponential Function
 Vertically stretched by a factor of 2
 Vertical shift of 9 units up
 Domain: {X ∈ Rx<0}
 Range: {Y ∈ R10<y<228.8}
Quadratic Function
 Vertically compressed by a factor of 4
 Reflected in the xaxis
 Horizontally shifted to the right by 12 units
 Vertically shifted up by 300 units
 Domain: {X ∈ R7.78<x<15.61}
 Range: {Y ∈ R}
Polynomial (Cubic) Function h=  (x24)^3 +152
 Sinusoidal Function h= 35 sin (t) +65
 Cosine Function h= 59.5 cos (t+17.18)+90

Polynomial (Cubic) Function
 Reflected in the xaxis
 Shifted to the right by 24 units
 Translated up by 152 units
 Domain: {X ∈ R}
 Range: {Y ∈ R151.1<y<52.812}
Sinusoidal Function
 Amplitude is 35
 Period is 2π
 Shifted up by 65 units
 Domain: {X ∈ R28.63<x<36.32}
 Range: {Y ∈ R}
Polynomial (Cubic) Function h= (t41.2)^3 +29
 Rational Function h=  30/(t43.8) +120
 Polynomial (Quartic) Function h = 1/6 (t65.8)^4 +30.1

Polynomial (Cubic) Function
 Shifted to the right by 41.2 units
 Vertical translation of 29 units up
 Domain: {X ∈ R}
 Range: {Y ∈ R100<y<30.74}
Rational Function
 Reflected in the xaxis
 Vertical asymptote at x=43.8
 Horizontal asymptote at y=120
 End Behaviour (x approaches +/ ∞)
 Domain: {X ∈ R45.3<x<61.004}
 Range: {Y ∈ R}
Exponential Function h=2^(t65.5)+28.7
 Quadratic Function h=  3.45 (t75.85)^2 +216.2
 Quadratic Function h= 2 (t87.12)^2 +12

Exponential Function
 Vertically stretched by a factor of 2
 Horizontal translation of 65.5 units to the right
 Shifted up by 28.7 units
 Domain: {X ∈ R}
 Range: {Y ∈ R30.1<y<125}
Quadratic Function
 Vertically compressed by a factor of 3.45
 Reflected in the xaxis
 Horizontal translation of 75.85 units to the right
 Vertical translation of 216.2 units up
 Domain: {X ∈ R}
 Range: {Y ∈ R125<y}
Linear Function h= 1/4 (t80) +15

All Equations: (with restrictions)
All Functions: (indicated on graph)
Calculations:
1.) Solve for the exact time(s) when the rollercoaster reaches a height of:
a) 250 feet
2.) Calculate Average Rate of Change from:
a) 10 to 15 seconds
3.) Calculate Instantaneous Rate of Change at: