Posted: February 17th, 2022





Culminating Task 



Course Name 












Design of the Roller coaster using DESMOS Online software

The equations and the variables 



A copy of the DESMOS graph 




Answers to the respective questions 

  1. The X-axis refers to the direction in which the roller coaster is moving in as shown by the advancement of DESMOS graphing. 
  2. The Y-axis refers to the vertical advancement of the rollercoaster, which could be either upwards or underground, represented by a negative connotation. 
  3. The polynomial equation is 

The parent equation for the above polynomial equation is 

  1. Considering the above equation 

The parent equation is multiplied by a and then added by 5, 

The transformation is thus described as follows 

  • Vertical stretch by a factor of a 
  • Vertical translation of 5 units up. 
  1. Considering the following polynomial 

, we have points (1,1) 

Impliying 1 =a 1^3 + 5 

This becomes 

1 =a + 5 , a =-4 

The transformed point is at a =-4

  1. The reasonable domain of a roller coaster is [0,200]

The range of the roller coaster is r(x) <=75

7. Considering the polynomial 

a is considered as 0.05 

Thus the polynomial becomes 0.05 

8. The largest change at the rollercoaster occurs at the curve

This is the curve where we experience the greatest change in the gradient during the movement within the roller coaster

9. The following is one of the trigonometric equations. 

The parent equation is y =sin x



On a scale of 1-10, I would rate my rollercoaster as 7 

My roller coaster is quite thrilling. Many sections would contribute to the thrill. The loops and the trigonometric transitioning would significantly contribute to the joy. 


To generate more excitement for the rollercoaster, I would add an underground transitioning to achieve more entertainment. 



The following data shows how the average salary of soccer players varies in respect to the value of their respective clubs. This implies that the more value an English premier league club is, the more likely it will pay its soccer players high wages, thus showing the equity for the data. 

Club Name  Club value in Billion The average salary of players  Yearly (in Million dollars ) 
Manchester United  4.65 7.66
Liverpool   4.14 6.92
Manchester City  4 8.73
Chelsea   3.35 5.97
Arsenal 3.06 5.99
Totten hum  1.7 4.95
West ham  0.804 3.78
Everton   0.518 5.13
Leicester City  0.237 4.19
Wolves   0.275 2.75


  1. For the club value, the scale is 1x 10^9

For the average salary of players, the scale is 1x 10^ 6

  1. The Graphing of the data 

4. The citation of the sources in APA 


English Premier League salary 2018/19. (n.d.). Statista.

Pompliano, J. (2021, May 20). The Most Valuable Premier League Clubs. Huddle Up.



The following is the modeling of the data using five respective functions 

  • Modeling in exponential form 

The modeled equation is given as 

  • Modeling in linear form

The linear equation used for modeling is

  • Modeling in Polynomial form 


The modeling equation is thus. 

  • Modeling using a power function 

The function used for modeling is 

Modeling using a logarithmic function 


The modeling equation is 

6. From the above modeling functions, the polynomial function modeled the data the best. This is mainly because many data points were close to the trend line, as shown by the graph 

7. The following equation shows the model as a single piecewise function 

F(x) = {

, 2.75≤x≤7.66}

8. from the following modeling using the polynomial function, we can effectively predict the correlation between the club value and the average wages they pay their soccer players. High valuable soccer players pay high salaries for their players compared to clubs with lower valuations. This trend is evident from the trend line. 

9. The model used for the data is reasonably accurate and effective. This is due to the number of data points that are close to the trend line of the data model. The trend line closely covers also half of the data points. 


Considering the data model, effective policies should thus be set up concerning the wages clubs should pay their soccer players. Clubs should therefore reconsider overpaying their players just because they can afford it. It is worth noting that these salary variations affect the performance of these respective clubs. To close the gap for competition, Clubs should conjointly discuss these wages with respective football organizations to minimize any disparity in the pay. 











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