Advanced Functions

Big image

Table of Contents

1. Summary of Creation of Graph

2. Rough Draft of Height vs. Time Graph

3. Final Copy of Height vs. Time Graph

4. General Information

5. Summary of Functions

6. Calculations

The Creation of the Rollercoaster

We wanted to create a rollercoaster that was, ideally, an enjoyable and realistic ride. Keeping in mind that we had to satisfy the criteria of including at least one linear, quadratic, polynomial of degree 3 or higher, rational, sinusoidal, exponential, and logarithmic function, we began playing around with different functions on paper before transferring it onto desmos, crafting together our rollercoaster. In the final copy of our height vs. time graph (pictured below), it is shown that our rollercoaster begins with a slight drop before elevating to prepare for and lead into the dramatic drop, which sets our ride apart from your average coaster. After the climactic drop, the acquired momentum is spread across a series of smaller drops and loops in order to allow the rollercoaster some time to slow down. As we had to ensure the height and time restraints were met, the main difficulty we encountered was the transferring of the functions from paper to desmos, so we had to experiment with transformations, specifically stretches and compressions. Some additional difficulties that we stumbled upon were making sure the different functions smoothly transitioned into one another, having to alter the domain of each function several times. Another obstacle that we came across was that we were unaware of the fact that the rollercoaster had to start and end at the same height, so we had to make subtle changes to some of the functions, expanding the domain and changing the 'c' value.

Height (ft) vs. Time (s) Graph - Rough Draft

Big image
Click title to view graph on Desmos.
Big image

General Information

Maximum Height: 250 ft

Minimum Height: 11.5 ft

Total Time: 100 s

Summary of Functions

Rational Function

Big image
  • Rational function with no zeroes

  • Shift up 11 ft

  • Horizontal asymptote at h=11 ft

  • No vertical asymptotes (negative discriminant when denominator is equal to 0)
Big image

Exponential Function

Big image
  • Exponential growth function (Base 2)

  • Shift up 11 ft

  • Horizontal asymptote at h=11 ft

Polynomial Function

Big image
  • Degree 3

  • Odd degree function with positive leading coefficient

  • The end behaviour is Q3-Q1
  • "Flattens" out around (11,60)

  • The point symmetry would be about the origin if the function was not shifted horizontally and vertically
  • Odd function, hence there is point symmetry about (11,60)
  • Odd function because h(-t) = -h(t):

h(-t) =(-t-11)^3 +60

h(-t) =-h(t)

Quadratic Function

Big image
Vertex form equation:
Big image
  • Downward opening parabola
  • The vertex is (26,250), due to a horizontal shift 26 units right and a shift up 250 ft.
  • The optimum value of the parabola is the maximum height of the entire rollercoaster (250 ft)

Logarithmic Function

Big image
  • Vertical stretch by a factor of 3

  • Reflection about the t-axis

  • Shift right by 41 units, shift up 16.6 ft

  • Vertical asymptote at t=41 due to shift right

Sinusoidal Function

Big image
Equivalent cosine equation:
Big image
  • Vertical compression by a factor of 1/2

  • Phase shift right 22 units, vertical shift up 13.5 ft

  • Period: 2π

  • Amplitude:0.5

  • Mid Axis: t=13.5

  • Maximum Point: 14 , Minimum Point: 13

  • # of cycles with domain: approximately 4.98 (4 full cycles + 6.17/2π cycle)

Linear Function

Big image
  • m=0 (constant speed)

Calculations

Time(s) when the rollercoaster is at a height of 12 ft:

Big image
Big image
Big image
Big image
Big image
Big image

Time(s) when the rollercoaster is at a height of 250 ft:

Big image
Big image
Big image
Big image
Big image
Big image
Big image
Big image

Instantaneous Rate of Change at 35 s:

Big image

Average Rate of Change from 10 to 15 s:

Big image

Average Rate of Change from 50 to 60 s:

Big image
Big image