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 tallest hill of the roller coaster  ride cannot exceed 180 feet if you choose to design a wooden roller  coaster.  If you choose to design a steel roller coaster, the tallest  hill cannot exceed 400 feet. 

–  The ride dives below the ground into a tunnel at least once. However,  the deepest tunnel cannot exceed 180 feet for either type of roller  coaster.

Complete the following tasks to write your report on your Roller  Coaster Design. Label each part clearly. Your work MUST be neat,  organized and must appear professional.

  1. Will your design be a wooden roller coaster or a steel roller coaster?
  2. Draw a rough sketch of your initial “roller coaster” ride on a  coordinate plane. Capture your graph in a jpeg or png image. It’s ok to  use your smartphone to capture the image; however, your the jpeg or png  formats in your solutions document. Note:  Be sure to illustrate and label your x-axis and y-axis scale to  identify the time length (in minutes) of the ride and the height of the  ride you are designing. 
  3. List all zero(s) of your polynomial as a coordinate pair; be sure to  include at least one of each of the following on your design: one  double root (multiplicity of two), at least 2 real distinct roots, and  imaginary roots. (Hint: It might be necessary to go back to your design and modify it according to these root requirements.) 
  4. Write the factored form of your roller coaster polynomial.
  5. Find the equation in standard form that represents your roller coaster ride.
  6. Perform long division and/or synthetic division to verify the correctness of your factors.
  7. Describe the end behavior of your function and give a reason why this makes sense for the roller coaster.
  8. Describe the behavior of the function at each x – intercept.
  9. State the practical domain of your graph (that is, the actual ride). Explain. Use complete sentences.
  10. State the practical range of your graph (that is, the actual ride). Explain. Use complete sentences.
  11. Specify the intervals where the polynomial is increasing, decreasing, or constant.
  12. Use Desmos Graphing Calclulator (Links to an external site.)  online to draw an accurate graph of your polynomial. Make sure to label  all zero(s) and y-intercept(s) exactly. Show the starting point of the  ride and the ending point of the ride. (Note that you do not have sign  up for an account to use the Desmos calculators.) 
  13. Use your graphing calculator to find the turning points. Round the  turning points to the nearest hundredth. Label the relative extrema as a  relative minimum or relative maximum.