# Roller Coaster Assignment

## Summary

I created my roller coaster by manipulating the parent equation of each function. For example: y= x² is the parent function for a quadratic. I knew that I needed to apply transformations to this particular function. So, to transform this function I had to change the “a” , “k”, “d” and “c” value of the quadratic to fit in the form of: a(kx-d)²+c. I essentially used vertical stretches / compressions horizontal stretches / compressions, vertical shifts: up or down and horizontal shifts to the left or right. I also applied reflections in the x and y axis to reflect certain functions in some parts of my roller coaster. Furthermore, it was important to apply certain restrictions to my functions and to essentially ignore the second, third and fourth quadrants (as they would make the roller coaster go underground and go backwards with negative height and time). One of the major difficulties I experienced when making my roller coaster was staying within the 100 second time limit. To overcome this difficulty, I had to alter the restrictions on my functions or change there domains to complete the ride in 100 seconds. Furthermore at the beginning, a difficulty I encountered was connecting the functions together to form the roller coaster. I had to use guess and check (or trial and error) to find the correct "a", "k" , "d" and "c" values. I also realized that I had to restrict the domain of my functions at certain times to create it.

## Written Report: Description of Functions of Height vs Time

I started my roller coaster from its minimum point. Therefore my y-intercept had to be (0, 10). So I essentially had to plug in zero (to the x value) of my first linear equation (y=20x+10) to see if the height would be 10. Moreover, this linear equation has a positive slope that is increasing, therefore as the time increases the height also increases. The domain of this roller coaster has to be x>0 as time cannot be negative or in this case the roller coaster cannot go backwards. The polynomial equation y= 2(x-5)³+100 also has a positive slope meaning that the roller coaster is accelerating rapidly as its height and time increases. The sinusoidal equation y= -5cos(x)+170 has a negative slope, thus the roller coaster is decelerating rapidly towards its trough, but then goes back up to its crest as it accelerates and gains speed. The quadratic equation y= -(x-27.5)²+300 reaches a maximum height at 300 feet. Then the roller coaster rapidly falls from it's maximum height as the time increases and picks up speed. The rational equation y= 2x²/2x-58 causes the height of the roller coaster to decrease, as time increases indicating the negative rate of change between two time intervals. The exponential equation y= -3^(x-80.3)+120 depicts exponential decay, meaning the height of the roller coaster is decreasing as time increases or in other words the ride is slowly coming to a stop. The logarithmic function y= -log(x-82)+100 indicates decay as well, as the height is slowly decreasing as time increases. Therefore, the speed of the roller coaster is also decreasing. The final linear equation y= -6.9x+700 has a negative slope indicating that the roller coaster is at its minimum height of 10 feet because the ride is over at 100 seconds.

## Equations of the Functions:

Polynomial Function

y= 2(x-5)³+100

{2≤x≤8.419}

Sinusoidal Function

y= -5cos(x)+170

{8.307≤x≤5.18pi}

y= -(x-27.5)²+300

{16.287≤x≤33}

Rational Function

y= 2x²/2x-58

{33≤x≤71}

Exponential Function

y= -3^(x-80.3)+120

{71≤x≤82}

Logarithmic Function

y= -log(x-82)+100

{82≤x≤87.06}

Linear Function

y= -6.9x+700

{87.06≤x≤100}

Linear Function

y=20x+10

{0≤x≤2.3}

## Roller coaster on Desmos:

The Link to my Roller Coaster on Desmos: