# The Dragon's Tale

### A New Designer Roller coaster

## The Creation of 'The Dragon's Tale"

For our roller coaster we began with a simple starting sketch, which only incorporated the maximum and minimum height needed,and the types of functions that are required in our coaster. We basically one by one started adding the things we needed for the coaster,

Problem 1:

- As we progressed forward in creating our roller coaster we noticed that we had to restrict the amount of time we give each incline and decline in out roller coaster. Because we forgot to incorporate the time limitation our coaster stopped at a level above ground. Therefore, the whole roller coaster ride ends above ground, which is impossible because to get off you needed to come back down to the ground (the minimum height).

So, in our second sketch we decided to add a time restriction and graph our coaster based on the amount of time each function gets. Which thoroughly helped us construct our roller coaster.

## Description of Functions of Height vs. Time

Transformations:

- a horizontal compression by a factor of 7.4
- translated 10 units up

Function 2: Max height-40.02 at 4.5 seconds

Transformations:

- a vertical reflection on the x-axis
- a vertical compression by a factor of 2
- shifted 9 units to the right
- translated 40.02 units up

Function 3: Max height- 37.4 at 5.34 seconds

Transformations:

- vertical reflection on x-axis
- vertical stretch by a factor of 2
- horizontal compression by a factor 2
- shifted 4.1 units to the right
- translated 36 units up

Function 4: Max height-59.84 at 8.12 seconds

Transformations:

- vertical stretch by a factor of 2
- horizontal compression by a factor of 2
- shifted 14 units to the right
- translated 37 units up

Function 5: Max height 73.49 at 19.97 seconds

Transformations:

- vertical stretch by a factor of 9
- horizontal compression by a factor of 2
- shifted 15.5 units to the right
- translated 61 units up

Function 6: Max height- 195.89 at 22 seconds

Transformations:

- vertical reflection on the x-axis
- horizontal asymptote at 22
- translated 73 units up

Transformations:

- a vertical reflection on x-axis
- a horizontal stretch by a factor of 1/2
- shifted 12.6 units to the right
- translated 209 units up

Function 8: Max height- 290.42 at 64.95 seconds

Transformations:

- vertical compression by a factor of 0.2
- horizontal compression by a factor of 2
- shifted 118 units to the right
- shifted 265.5 units up

Function 9: Max height- 209

Transformations:

- a vertical stretch by a factor of 2
- horizontal compression by a factor of 2
- shifted 26.1 united to the right
- translated 203 units up

Function 10: Max height- 265.6 at 59.6 seconds

Transformations:

- a horizontal compression by a factor of 3
- shifted 106.8 units to the left
- translated 20 down

Function 11: Max height- 300 at 68 seconds

Transformation:

- vertical reflection of the x-axis
- shifted 68 units to the right
- translated 300 units up

Function 12: Max height- 104.79 at 81.9 seconds

Transformations:

- vertical reflection on the x-axis
- vertical compression by a factor of 2
- shifted 164.8 units to the right
- translated 103.5 units up

Function 13: Max height-70

Transformations:

- vertical stretch by a factor of 10
- shifted 87.9 units to the right
- translated 60 units up

Function 14: Max height- 67.55

Transformations:

- vertical reflection on the x-axis
- horizontal stretch by a factor of -1/5
- horizontal reflection of the y-axis
- shifted 18 units to the left
- translated 20 units up

## The Equation of Each Function

__linear function__which had the equation:

**y=7.4x+10**with a time restriction of

**{0<x<3.76374586}**

2. The second function was a __quadratic function__ that had the equation:

**y=-(2x-9)^2+40.02** with a time restriction of** {0<x<3.76374586}**

3.The third function was a __sinusoidal function__ with the equation:**y=-2 sin (3x-4.1)+36** with a time restriction of **{5.3363905936<x<6.6051017516* 10^0}**

4. The fourth function was a__ cubic function__ with the equation: **y=2(2x-14)^3+37,** with a time restriction** {6.6051017516* 10^0<x<8.12686906355* 10^0}**

5.The fifth function was a __logarithmic function __with the equation: **y=9 log (2x-15.5)+61** with the time restriction of **{8.13<x<19.97}**

6. The sixth function was a__ rational function __with the equation:**y=-(1/x-22)+73 **with the time restriction of **{19.9661394<x<21.991854} **and a height restriction of {72< y<200}

7. The seventh function was a __quartic function __with the equation:y=**-2(1/2x-12.6)^4+209** with a time restriction of **{21.991854<x<27.93708}**

8. The eighth function was an __exponential function__ with the equation:

**y= 0.2* 1.5^(2x-118)+265.5 **and a time restriction of **{59.683<x<64.9696}**

9. The ninth equation was a __sinusoidal function__, with the equation:

**y= 2 sin(2(x-26.1))+203 **and a time restriction of**{27.93702<x<38.4312}**

10. The tenth function was a __linear function__ with the equation: **y=(3x+106.8)-20 **with a time restriction of **{38.4312<x<59.683}**

11. The eleventh function was a __quadratic function__ with the equation: **y=- (x-68)^2+300 **with a time restriction of** {64.957<x<82}**

12. The twentieth function was a __cubic function__ with the equation:

**y=-(2x-164.8)^3+103.5** and a time restriction of **{82<x<84.068}{55<y< 109.3}**

13. The thirteenth function was an __sinusoidal function__, with the equation:

**y=10 sin (x-87.9)+60**, and a time restriction of **{84.0681<x<90.13}**

14. The last function was a __logarithmic function__ with the equation:

**y=-30 log (-(-1/5x+18))+20** and a time restriction of **{90.126<x<100}**

## What are the Times When the Roller coaster is at 250 feet?

the equations are:

y=(3x+106.8)-20 and **y=- (x-68)^2+300**** **

To solve for the first time the roller coaster reaches 250 feet you would use the linear equation.

__Step 1:__ Plug 250 into y

250=(3x+106.8)-20

__Step 2:__ Add 20 to each side

270=(3x+106.8)

__Step 3:__ Subtract 106.8 from each side

163.2 =3x

__Step 4:__ Divide each side by 3

54.4=x

Therefore 54.4 seconds is the first time the roller coaster reaches 250 feet.

To solve for the second time that the roller coaster reaches 250 feet, you would have to use the quadratic function equation.

__Step 1:__ Plug in 250 as y

250=-(x-68)^2+300

__Step 2:__ Subtract 300 from both sides

-50=-(x-68)^2

__Step 3:__ Divide both sides by negative 1

50=(x-68)^2

__Step 4: __Square root both sides

7.07=x-68

__Step 5: __Add 68 to both sides

75.07=x

Therefore at 75.07 seconds the roller coaster is at 250 feet again.

## But, What are the Times When the Roller coaster is at 12 feet?

y=7.4x+10 and y=-30 log (-(-1/5x+18))+20

To determine the first time the roller coaster reaches 12 feet you would use the first equation.

y=7.4x+10

__Step 1:__ Plug in 12 as y

12=7.4x+10

__Step 2__: Subtract 10 from both sides

2=7.4x

__Step 3:__ Divide both sides by 7.4

0.270=x

Therefore at 0.270 seconds the roller coaster is at 12 feet for the first time.

To determine when the roller coaster is at 12 feet for the second time, you would use the second equation

**y=-30 log (-(-1/5x+18))+20**

__Step 1: __Plug in 12 as y

12=**y=-30 log (-(-1/5x+18))+20**

__Step 2:__ Subtract 20 on both sides

-8=**-30 log (-(-1/5x+18))**

__Step 3:__ divide both sides by -30

0.266=** log (-(-1/5x+18))**

__Step 4:__ Log inverse both sides

1.845=** (-(-1/5x+18))**

__Step 5:__ Add 18 to both side because we expanded the outer negative 1 into the brackets

19.845=1/5x

__Step 6: __Multiply both sides by 5 and divide both sides by 1

99.2=x

Therefore, at 99.2 seconds the roller coaster is at 12 feet for the second time.

## Average Rate of Change from 10 seconds to 15 seconds and 50 to 60 seconds

To calculate we will need to use two points from the function on the graph, the points that we will use are (10,66.88) and (15,71.45)

Step 1: Get the average rate of change equation

y2-y1/x2-x1

Step 2: Plug in the respective values for each variable

71.45-66.88/15-10

Step 3: Solve for the the rate of change

4.57/5

rate of change=0.914

**Therefore the rate of change is 0.914 feet/ second**

To calculate the average rate of change for 50 to 60 seconds you would need the points:

(50,236.8) and (60,265.95)

Step 1: Use ROC equation

y2-y1/x2-x1

**Step 2: Plug in the values of each variable**

265.95-236.8/60-50

Step 3: **Solve for ROC**

ROC=29.15/10

ROC=2.915

**Therefore the average rate of change from 50 seconds to 60 seconds is 2.9 feet/second**

## Instantaneous Rate of Change at 35 seconds

To calculate the instantaneous rate of chan using the equation:

y= 2sin(2(x-26.1))+203 (Using this equation find the y-value

POINTS: (35,203.3)and (35.001,203.3106)

Step 1: Use the equation for IROC

y2-y2/x2/x1

Step 2: Plug in the values for each variable

203.3106-203.3/35.001-35

Step 3: Solve for the IROC

IROC=10.6 feet/second

Therefore the Instantaneous rate of change is 10.6 feet/ second.