# Advanced Functions

## 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

## 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)