The Sharmanator

Rollercoaster Summative

By: Krupali, Krishna, Darren and Khalid

Summary

The first steps our group took to start the construction of our rollercoaster was by writing the base equations down for each function. This made it easy to distinguish between functions, and to lay out which function would go where. Next, we drew out a rough graph of our rollercoaster and started labeling each function. We also made sure our functions did not exceed the maximum height of 300. We first decided to use a linear equation, since it was the easiest function to graph, and it would create a good starting point for us. We made our linear function approach our max value of 300. Secondly, we used a sinusoidal function, since it starts after the linear at the max point, it was easier to use a cosine function. Then we included an inverse quadratic function as our second drop. A rational function starts to bring the roller coaster back up, for another drop. We used a polynomial to the fourth degree for a wider drop. Then, we decided to include a loop in our rollercoaster, we started off the loop with a log function, and then used another sin function to make the last drop. We brought the coaster to an end with a second log function, to a full 100 seconds, and ended at the point where it started (max- 10m). Some difficulties our group faced were fitting all these functions in the actual time span of 100 seconds without making it look like one big mess. Also, making sure the functions intersected at points was difficult to do.

Height vs Time Graph

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Height vs Distance Graph

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General Descriptions of Functions:

Linear Function:

y= 30x+10 (0 ≤ x ≤ 9.66}

Ø Y- intercept a (0, 10)

  • Represented by ‘b’ value in y=mx+b

Ø Constant Rate of Change 30ft

  • Represented by “m” value in y=mx+b

Ø Function runs from 0 to 9.66 seconds

  • Domain { XER| 0≤ x≤ 9.66}

Ø Function goes from 10ft to 300ft

  • Range {YER| 10≤ y≤ 300}

Ø Increasing Function

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Cosine Function:

y= 100 Cos [0.19(x-10)] + 200 {9.66≤ x≤ 30}

Ø Amplitude is 100

  • Represented by ‘a’ value in y= acos [k(x-r)] +h

Ø Middle axis is at y= 200

  • Represented by ‘h’ value in y=acos [k(x-r)]+h

Ø Function runs from 9.66 to 30 seconds

  • Domain {XER| 9.66≤ x≤ 30}

Ø Function goes from 100ft to 300 ft

  • Range: {YER| 100≤ y≤ 300}

Ø Transformations:

  • Vertical stretch by a factor of 100
  • Horizontal Stretch by a factor of 1/0.19
  • Shift 10 units to the right
  • Shift 200 units up
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Quadratic Function:

y= -2(x-38)^2 + 250 {30≤x≤ 45}

Ø Parabola opens downwards

Ø Vertex at (38, 250)

Ø AOS at x= 38

  • Represented by the ‘h’ value in y= a(x-h)^2+k

Ø Optimal value at y=250

  • Represented by the ‘k’ value in y= a(x-h)^2+k

Ø Decreasing rate of change

Ø Function runs from 30 to 45 seconds

  • Domain {XER|30≤ x ≤ 45}

Ø Function goes from 122ft to 250ft upwards

  • Range {YER|122 ≤ y ≤ 250}

Ø Transformations:

  • Vertical Reflection
  • Vertical Stretch by a factor of 2
  • Shifted 38 units to the right
  • Shifted 250 units up
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Exponential Function:

y= 2.33^ -2(x- 47.3) + 105

Ø Horizontal Asymptote at y= 105

  • Represented by the ‘k’ value in y= b^(x-r) +k

Ø Function goes from 45 to 55 seconds

  • Domain: {XER| 45≤ x≤ 55}

Ø Function goes from 154 ft to 105 ft

  • Range: {YER| 105 ≤ y≤ 154}

Ø Exponentially decreasing function

Ø Transformations:

  • Horizontal Reflection
  • Horizontal compression by a factor of 1/2
  • Shifted 47.3 to the right
  • Shifted 105 units up
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Rational Function:


y= -54.75/x-62.3 + 97.5 {55≤ x≤ 61.13}

Ø Vertical Asymptote at x= 62.3

  • Represented by ‘r’ value in y= a/x-r +h

Ø Horizontal Asymptote at y= 97.5

  • Represented by ‘h’ value in y= a/x-r +h

Ø Function runs from 55 to 61.13 seconds

  • Domain: {XER| 55≤x ≤61.13}

Ø Function runs from 143.1ft to 105ft

Range: {YER|105≤ y ≤ 143.1}

Ø Transformation:

  • Vertical reflection
  • Vertical stretch by a factor of 54.75
  • 62.3 units to the right
  • 97.5 units up
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Polynomial Function:

y= -0.2 (x-65.98)^4 + 255 {61.13 ≤ x≤ 71.5}

Ø End Behaviour: Q3- Q4 (for unrestricted domain)

  • Negative leading coefficient/ ‘a’ value in y=a (x-h)^4 + k

Ø Function runs from 61.13 seconds to 71.5 seconds

  • Domain: {XER| 61.13 ≤x ≤ 71.5}

Ø Function goes from 74.6ft to 255ft

  • Range: {YER| 74.6 ≤y ≤ 255}

Ø Transformations:

  • Reflection in x-axis
  • Vertical compression by a factor of 0.2
  • Shift 65.98 to the right
  • Shift 255 units up
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Log Function

y= -45log (x-70) +80 {71.5≤ x≤ 74.75}

Ø Inverse of an exponential function

Ø Vertical Asymptote at x= 70

  • Represented by ‘r’ value in y=-alog (x-r)+h

Ø Function runs from 71.5 to 74.75 seconds

  • Domain: {XER| 71.5 ≤x ≤ 74.75}

Ø Function goes from 72.1ft to 49.8ft

  • Range:{YER| 49.8 ≤ y ≤ 72.1}

Ø Transformations:

  • Vertical Reflection
  • Vertical Stretch by a factor of 45
  • Shift 70 units right
  • Shift 80 units up
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Sinusoidal Function:

y= -30Sin[0.4(x-8)] + 80 {74.75 ≤ x≤ 89.5}

Ø Amplitude is -30

  • Represented by ‘a’ value in y= asin [k(x-r)]+h

Ø Middle axis is y=80

  • Represented by ‘h’ value in y= asin [k(x-r)]+h

Ø Function runs from 74.75 to 89.5 seconds

  • Domain: {XER| 74.75 ≤x ≤ 89.5}

Ø Function goes from 50ft to 110 ft

  • Range: {YER| 50 ≤ y≤ 110}

Ø Transformations:

  • Reflection in the x axis
  • Vertically stretched by a factor of 30
  • Horizontally stretched by a factor of 5/2
  • Shift 8 units to the right
  • Shift 80 units up
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Logarithm Function:

-45log (x- 88) + 57 {89.2 ≤x≤100}

Ø Inverse of an exponential function

Ø V.A at x=88

  • Represented by the ‘r’ value in y= alog (x-r)+h

Ø Function runs from 89.2 to 100 seconds

  • Domain: {XER| 89.2≤ x≤ 100}

Ø Function goes from 52.6ft to 30ft

  • Range: {YER| 30≤ y≤ 52.6}

Ø Transformations:

  • Reflection in the x-axis
  • Vertical Stretch by a factor of 45
  • Shift by 88 units right
  • Shift by 57 units up
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Calculations:

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Finding Equations

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Rough Graphs

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