# The Functional Death

## CONCEPTUAL MINDS INCORPORATED

"Conceptual Minds Inc. " is a roller coaster design company based in Toronto, Ontario. The company has completed several roller coaster projects all over the world. Their most recent project was The Functional Death at Paradox Point which is the largest amusement Park in Ontario.

## "The Functional Death" is now ready to run

The long awaited thrill filled ride, The Functional Death has now opened to the public. You can come to Paradox Point to enjoy the spine chilling experience drop from 300 ft!

## Specifications

Type: Steel

Launch System: Hydraulic Propulsion

Maximum Height: 300 ft

Minimum Height: 10 ft

Maximum Speed: 151 km/h

## Creation of Roller Coaster

Before the construction of the roller coaster, our team of engineers, designed the roller coaster by using mathematical functions! The team utilized a variety of functions to graph the roller coaster. This flyer will describe the process of creating the roller coaster, and all of the math used.

1. Rough Plan

2. Summary of Plan

3. General Descriptions of Equations

4. Time Calculations

5. Average/Instantaneous Rate of Change Calculations

6. Gallery of The Functional Death

## Plan For Construction of Graph

Constructing The Functional Death involved several steps, from planning to mathematical substitution. First, we started by sketching a blueprint of our roller-coaster on paper, and labelled where we could place each type of function according to the appearance of the segment. In this blueprint, we also decided approximately how much time each segment of the ride would last. Subsequently, we began constructing our roller-coaster on Desmos. In order to obtain the appropriate horizontal translations, we utilized the x-restriction value of the previous function. Moreover, in order to find the vertical translations we used the y-value of the last point on the previous function, and the y-value of the first point on the new function. We found the difference between these two values and used it to find the exact c-value. Furthermore, in order to find the exact a-value of the function, we substituted the last point of the previous function into the equation we created using the c and d values. We made our transformations as accurate as possible by adding more decimals and this was done through zooming extensively into the graph on Desmos. The main difficulty we faced was making functions perfectly align in sequence, as it was difficult to find the exact translations and a-values. Another difficulty we encountered was making the roller-coaster flow smoothly without sharp edges.

## General Descriptions of Equations

The roller-coaster we have created consists of multiple different kinds of functions. A function is a relation for which each value from the first set is associated with exactly one value from the second set. In other words, each x value is paired with exactly one y value. Since we have created a Height vs. Time graph, the x-axis is time in seconds, and the y-axis is height above the ground in feet.

Here is a breakdown of each different kind of function utilized:

Linear: A linear function is a degree one polynomial function . The equation produces a straight line when plotted on a graph. A linear is found in the form y= mx + b. Where "m" is the slope of the line (rise/run) and "b" is the y- intercept.

A quadratic function is a degree two polynomial function. A quadratic can be in many forms which include (Standard Form, Vertex Form and Factored Form). One of the common forms is vertex form, which is expressed as y = a [k(x - d)]^ 2 + c.

## Polynomial:

A polynomial function is a function such as a quadratic, a cubic, a quartic, and so on, consisting of only positive integer powers of x. For simplicity we used the vertex form rather than the standard form.

## Rational:

Rational functions are a ratio of two polynomials. A rational function is found in the form y= a/x-b +c. When the (x-b) equals zero, it creates a vertical asymptote. The value of the vertical asymptote is equal to the value of x that causes the denominator to be zero.

## Sinusoidal:

A sinusoidal function is a repetition of oscillations. A sinusoidal function includes the three trigonometric functions which are sine, cosine, and tangent. In our roller coaster we used the sine and cosine functions. A sine function is found in the form

y = a sin [k(x – d)] + c. A cosine function is found in the form y = a cos [k(x – d)] + c.

The k value is determined by 2pi/Period.

## Exponential:

An exponential function is a constant raised to the power of the argument, where a and b are greater than 0, and b is not equal to 1. An exponential is found in the form y = ab^ [k(x-d)] + c.

## Logarithmic:

A log function is the inverse of an exponential function. A log function is found in the form y = a log [k(x – d)] + c. A base logarithmic function has a vertical asymptote at x=0.

## 3-D Model of The Functional Death

This is the 3-D Model of The Functional Death. This model was utilized as a reference during the construction of the roller-coaster.