Advanced Functions

Table of Contents
1. Summary of Creation of Graph
3. Final Copy of Height vs. Time Graph
4. General Information
5. Summary of Functions
6. Calculations
The Creation of the Rollercoaster
Height (ft) vs. Time (s) Graph - Rough Draft


General Information
Minimum Height: 11.5 ft
Total Time: 100 s
Summary of Functions
Rational Function

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)

Exponential Function

Exponential growth function (Base 2)
Shift up 11 ft
- Horizontal asymptote at h=11 ft
Polynomial Function

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


- 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

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


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

- m=0 (constant speed)
Calculations
Time(s) when the rollercoaster is at a height of 12 ft:






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








Instantaneous Rate of Change at 35 s:

Average Rate of Change from 10 to 15 s:

Average Rate of Change from 50 to 60 s:

