# "Geometric Cyclone"

## Expectations of the Assignment

In this assignment we were expected to create a roller coaster using these functions:

1 linear

 1 polynomial

 1 rational
 1 sinusoidal
 1 exponential

 1 logarithmic

We were required to make this on a graphing software such as "Desmos." We had to follow the guideline of making the ride 100 seconds only and the maximum height of 300ft and minimum height of 10ft above the ground.

## Summary of our Design Process

The plan we had for the designing a rollercoaster assignment was well organized and followed. As we started we brainstormed many ideas about what equations we need to have and where they will be placed. We also thought about which functions would be best fit for each part of the rollercoaster. An example is the 4th degree function we used at 20.5 seconds till 55 seconds which was at the maximum height of 300ft was a best fitted in that situation because it was used as the drop. Also another example is the linear function we used at 70 seconds to 80 seconds was also an appropriate fit for the rollercoaster because it showed the rollercoaster slowing down and getting closer to its minimum height. We decided how many seconds each function will take and divided the rollercoaster accordingly to the timing we were given. Through this assignment our group faced difficulties while creating the rollercoaster. The difficulties we faced were trying to make all the equations fit together and not make it seem congested. It was also difficult deciding where to place each function so it would look like a rollercoaster. The time that was given which was 100 second was also hard to manage as we had to set out specific timings for each equation we made to meet the time requirement. It was also challenging when we had to guess and check the vertical shift, horizontal shift, stretches and compressions to make sure they fit according to the rollercoaster. When we organized what we wanted to do it made it easier for us to create the rollercoaster.

## The Functions Used

1. Linear Function: y=80, y=-x+144, y=-2x+218.12, y=85.7

2. Quadratic Function: y= (2x-13)^22 +10

3. Log Function: y=7log(x-10)+81.1

4. Sinusoidal Function: y= 8cos(x-15) +78, y=-14sin(0.5x-76)+72

5. Exponential Function: y=6^-(x-56)+85.99, y=6^(x-24)+85.99

6. Rational Function: y= 0.8/x-63.6 +78

7. Polynomial Function: y= 1(0.25x-10) ^4 +300

## Linear Function: y=80

• Linear Function
• Horizontal Line
• Restrictions: D= {XER/0 ≤ x ≤ 2.316};
• Time has to be greater OR equal to 0
• Time has to be less than OR equal to 2.316
• Restrictions: R={YER/y=80}
• Y-Intercept at y=80
• Slope is 0

## Quadratic Function: y= (2x-13)^2 +10

• Opens up parabola
• Restrictions: D={XER/2.32≤ x ≤ 10.68}
• Time has to be greater than OR equal to 2.32
• Time has to be less than OR equal to 10.68
• Restrictions: R= {YER/y≤80)
• Horizontal compression by a factor of 0.5
• Horizontal translation by 6.5 units to the right
• Vertical translation upwards by 10 feet upwards
• Before the vertex, function is decreasing and negative slope
• After the vertex, function is increasing and positive slope
• Vertex is (6.5,10)
• Where 6.5 represents the time
• Where 10 represents the height in feet

## Log Function: y=7log(x-10)+81.1

• Log function
• Restrictions: D={XER/15 ≥ x ≥ 10.69}
• Time has to be less than OR equal to 15
• Time has to be greater than OR equal to 10.69
• Restrictions: R={YER/80.02 ≤ y ≤ 85.99}
• Height has to be greater than OR equal to 80.02
• Height has to be less than OR equal to 85.99
• Vertical stretch by a factor of 7
• Function is increasing with a positive slope
• Horizontal translation by 10 units to the right
• Vertical translation by 81.1 feet upwards

## Sinusoidal Function: y= 8cos(x-15) +78

• Restrictions: D= {XER/21.283 ≥ x ≥ 15};
• Time has to be less than OR equal to 21.283
• Time has to be greater than OR equal to 15
• Restrictions: R= {YER/ y ≤ 86}
• Height has to be less than OR equal to 86 Amplitude is 8
• Cosine function
• Amplitude is 8
• Phase shift to the right by 15 units
• Axis of Symmetry is 78

## Exponential Function: y=6^(x-24)+85.99

• Restrictions: D={XER/24.8977 ≥ x ≥ 21.283}
• Time has to be less than OR equal to 24.8977
• Time has to be greater than or equal to 21.283
• Restrictions: R= {XER/86 ≤ y ≤ 90.39}
• Height has to be greater than OR equal to 86
• Height has to be less than OR equal to 90.39
• Base is 6
• Horizontal translation right by 24 units
• Vertical translation upwards by 85.99 feet

## Polynomial Function: y= (0.25x-10)^ 4 +300

• Restrictions: D= {XER/ x ≥ 90.39}
• Time has to be greater than OR equal to 90.39
• Restrictions: R= {YER/90 ≤ y}
• Height has to be greater than OR equal to 90
• Quartic function (4th degree function)
• Reflection n x-axis
• Horizontal stretch by a factor of 4
• Horizontal translation by 40 units
• Vertical translation by 300 feet

## Exponential Function: y=6^-(x-56)+85.99

• Restrictions: D= {XER/ x ≤ 58}
• Time has to be less than OR equal to 58
• Restrictions: D= {YER/y ≤ 90}
• Height has to be less than OR equal to 90
• Base is 6
• Reflection in the y-axis
• Horizontal translation 56 units the right
• Vertical translation 85.99 feet upwards

## Rational Function: y= 0.8/x-63.6 +78

• Restrictions D={XER/64≤X≤70}
• Time has to be greater than or equal to 64
• Time has to be less than or equal to 70
• Restrictions R={YER/Y≤80}
• Height has to be less than or equal to 80
• Rational function
• vertical asymptote at x=63.6 sec
• Decreasing function towards +70
• slope is positive and decreasing towards the +ve infinity

## Linear Function: y=-x+144

• Restrictions R={YER/80≤Y≤86}
• Height has to be greater than or equal to 80
• Height has to be less than or equal to 86
• Linear function
• Decreasing function
• Slope is -1

## Linear Function: y=-2x+218.12

• Restrictions D={XER/70≤X≤80}
• Time has to be greater than or equal to 70
• Time has to be less than or equal to 80
• Restrictions R={YER/58.12≤Y≤78.12}
• Height has to be greater than or equal to 58.12
• Height has to be less than or equal to 78.12
• Linear function
• Decreasing function
• Slope is -2

## Sinusoidal Function: y=-14sin(0.5x-76)+72

• Restrictions D={XER/80≤X≤99}
• Time has to be greater than or equal to 80
• Time has to be less than or equal to 99
• Restrictions R={YER/58≤Y≤86}
• Height has to be greater than or equal to 58
• Heght has to be less than or equal to 86
• Amplitude is 14
• Phase shift to the right by 152 feet
• K value is 0.5
• Axis of symmetry is 72 feet up

## Linear Function: y=85.7

• Restrictions D={XER/99≤X≤100}
• Time has to be greater than or equal to 99
• Time has to be less than or equal to 100
• Linear function
• Horizontal line

## Calculations: Time at 250 feet

Time at 250 feet

250= -(1/4x - 10)^4 +300

-50= -(1/4x - 10)^4

50= (1/4x - 10)^4

50= [1/4(x-40)]^4

50= [(1/4)^4(x-40)^4]

50= 1/256 (x-40)^4

12800= (x-40)^4

+/-\sqrt[4]{12800}= x-40

+/-\sqrt[4]{12800} + 40= x

\sqrt[4]{12800} + 40= x

x= 10.63 + 40

x= 50.63 s

OR

- \sqrt[4]{12800} + 40= x

x= -10.63 + 40

x= 29.36 s

## Calculations: Time at 12 feet

12= (2x - 13)^2 + 10

12-10= [2(x=13/2)]^2

2= 4(x-13/2)^2

2/4= (x-13/2)^2

+/-\sqrt 1/2= x - 13/2

+/-\sqrt 1/2 + 13/2= x

\sqrt 1/2 + 13/2= x

x= 7.207 s

OR

-\sqrt 1/2 + 13/2= x

x= 5.793 s

## Calculations: Average Rate of Change from 10 to 15 sec

AROC= f(x2) - f(x1)/(x2 - x1)

= f(15) - f(10)/(15 - 10)

= [8cos(x-15) + 78] - [(2x - 13)^2 +10]/5

= [8cos(15-15) + 78] - [(2(15) - 13)^2 +10]/5

= (86 - 59)/5

= 5.4 ft/sec [going up]

## Calculations: Average Rate of Change from 50 to 60 sec

AROC= f(x2) - f(x1)/(x2 - x1)

= f(60) - f(50)/(60 - 50)

= [-x + 144] - [-(1/4x - 10)^4 + 300

= [-60 + 144] - [-(1/4(50) - 10)^4 + 300

= (84 - 260.94)/10

= -17.694 ft/sec

## Calculations: Instantaneous Rate of Change from 35 Seconds

35.001 & 35

IROC= f(35.001) - f(35)/35.001 - 35

= [-(1/4(35.001) -10)^4 +300] - [-(1/4(35) - 10]^4 +300]/0.001

= (247.5605 - 297.56)/0.001

=0.5 ft/sec