Roller Coaster Model - Hydrophius

Summative Task

Final Rough Draft

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Desmos Final Copy

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Functions on the graph

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Summary

Before equations of the rollercoaster had been found, we had to decide on what the rollercoaster would actually look like. All the members of the group had made a rough draft of what they believe the rollercoaster design would be. Afterwards we had met up and decided on what the final design of the coaster would be. With the finalized design we went on to graph the rollercoaster, at first using the base function of the functions needed and then adding the transformations and restrictions to get the final equations of the rollercoaster. The one major difficulty that we had was that we had some trouble figuring out the equations of all the functions that we had used in the rollercoaster. This was difficult because it took some time to get the equations that would graph the rollercoaster exactly to what we had on the rough draft. Additionally, another difficulty that we faced was when we had to decide on the final design. This was because we had four different graphs that we had to decide between, to fix this problem we picked the best two graphs and then tried to graph both of them to see which one would look better on Desmos.

Written Report

In a height vs. time graph, we know our dependent variable is height ( y-axis) and our independent variable is time (x-axis). Each of the following functions: Linear, Quadratic, Polynomial, Rational, Sinusoidal, Exponential and Logarithmic, can be described based on height vs time graphs.


  • Linear functions are written in the form of y=mx + b and when graphed, all the (x,y) points that are found, all lie on a straight line.
  • Quadratic functions are written in the form of y=a[k(x-d)]²+c.
  • Polynomial functions are functions which can be written in quadratic, cubic, quartic and so on.
  • Rational functions are functions with ratios of 2 polynomials. So the numerator can be divided by the denominator.
  • Sinusoidal functions are symmetrical waves which repeat at regular intervals. The correct form of writing sinusoidal functions is--> f(x)=a sin/cos[k(x-d)]+c.
  • Exponential functions are written in the form of y=ab^x where b>0, and b cannot equal 0.
  • Logarithmic functions are written in the form of y=alog[k(x-d)]+c


Below you will see the equations used to make the roller coaster categorized based on their type of function, along with their restriction on time.
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Calculations

Solving for exact time when a roller coaster reaches a certain time.

(a) Exact time at 250 feet when a roller coaster reaches a certain time(s).

(b) Exact time at 12 feet when a roller coaster reaches a certain time(s).

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Calculating the Average Rate of Change at a given time.

(a) Average Rate of Change from 10 to 15 seconds.

(b) Average Rate of Change from 50 to 60 seconds.

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Therefore, the average rate of change from 10 to 15 seconds is -5 and the average rate of change at 50 to 60 seconds is 3.79.

Calculating the instantaneous rate of change at a given time

(a) 35 seconds
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Therefore, instantaneous rate of change at 35 seconds is 0.32