# Roller Coaster Model - Hydrophius

## 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.

## 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).

## 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.

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
Therefore, instantaneous rate of change at 35 seconds is 0.32