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Grace Thomas: Roller Coaster Polynomials
1,338
Roller Coaster Polynomials
Factored Form
y= a (x-2) (x-2) (x-3) (x-4) (x^2+4)
y= a (x^6 -11x^5 + 48x^4 -120x^3 +224x^2 -304x +192)
Maxima and Minima
Y-Axis (Height)
(0,250)
This sketch is similar, but not exact to the digital graph. Below in green were the intended maxima and in blue were the intended minima.
Maxima: (2.6, 2.9)
Minimas: (2,0) (3.676, -14.00)
(4,0)
(3,0)
(2,0)
y= (1.3x^6 -14.3x^5 + 62.4x^4 -156x^3 +291x^2 -395x +249.6)
X-Axis (Time)
Y-Axis (Height)
y= 1.3 (x^6 -11x^5 + 48x^4 -120x^3 +224x^2 -304x +192)
(0,250)
250= 192a
1.3= a
(2,0)
Standard Form
(3,0)
(4,0)
Intercepts
X-Axis (Time)
Zeros:
x= (2,0)
x= (2,0)
x= (3,0)
x= (4,0)
x= (x^2+4)
Y intercept:
y= (0,250)
Original Sketch
Y-Axis (Height)
(0,250)
(2,0)
Synthetic Division
(3,0)
(4,0)
This coaster begins at 250 feet and the ride lasts 4 minutes long. It dips down beneath the surface at approximately 3 minutes.
X-Axis (Time)
Because the remainders are zero, the x-intercepts work. To confirm the equation, graphing will need to be done.
Task and Requirements
You have decided to become a structural engineer who specializes in roller coaster design. Your job is to design your own roller coaster ride. To complete this task, please follow these guidelines given to you by the amusement park:
- Your roller coaster ride must have at least 3 relative maxima and/or minima.
- The ride length must be at least 4 minutes.
- The roller coaster ride starts at 250 feet.
- The ride dives below the ground into a tunnel at least once.
x= 0
y= 250
x= 4
y= 0
End Behavior
Domain and Range
Y-Axis (Height)
Increasing and Decreasing Intervals
(0,250)
Domain: 4> x > 0
Range: 250> y > -14
Mathematically, this coaster would have infinite endings. Considering this is a roller coaster, the in creasing and decreasing intervals would begin at 250 feet and then end at the x-axis around 4 minutes.
(3,0)
(4,0)
X-Axis (Time)
Red= Increasing
(0,2) (2.6, 3.7)
X-Axis (Time)
Blue= Increasing
(2, 2.6) (3.7, 4)
The range is based off of the y-axis, which is height. Because the ride begins at a maximum height of 250 and then dips below surface to -14, it can not be greater or less than those values.
(4,0)
Y-Axis (Height)
(0,250)
The domain is based off of the x-axis, which is time. Because the ride has a duration of 4 minutes, it can not exceed that limit or be less than 0 minutes long.
(3,0)
Algebra II
Block 4B
Ms. Margenau
December 2016
made by
Grace Thomas
grace.thomas@msdk12.net