PART $2$: ROLLER COASTER DESIGN $(64$ total points$)$

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 the steps below.

The amusement park you are designing for gave you the following coaster requirements:

$-$ your coaster ride must have at least $3$ relative maxima and$/$or minima

$-$ the ride length must be at least $90$ seconds

$-$ the coaster ride starts at least $250$ feet above ground

$-$ the ride dives below the ground $(\backslash ($ x $\backslash )-$axis$)$ into an underground tunnel at least once

$-$roots must include one double root, at least two real roots $($rational or irrational$),$ and at least two imaginary roots

Either type or neatly handwrite your report on your Roller Coaster Design. Do NOT put your work here. Graph your function using Desmos or another graphing application. Label each part clearly. Your work MUST be neat, organized and appear professional.

$1.$ Create a function, $\backslash ($ R$($x$)\backslash ),$ using the above criteria. List all zeros $($real and imaginary$)$ of your polynomial.

$2.$ Write the complete factored form of your roller coaster polynomial.

$3.$ Describe the end behavior of your function and give a reason for this behavior.

$4.$ Use your calculator to find the times and heights at which each of the maximum and minimums of your rollercoaster occur.

$5.$ Create an accurate sketch of your polynomial graph using Desmos or another graphing application, making sure to label all axes appropriately.

$6.$ State the practical domain and range of your graph $($that is$,$ the actual ride$),$ including the length $($in time$)$ of your rollercoaster ride.

$7.$ Color the graph blue where the polynomial is increasing and red where the polynomial is decreasing and identify intervals of increasing, decreasing, and constant.

$8.$ List the intervals on which your polynomial function is positive and the intervals on which your polynomial function is negative.

Written Requirement: Write a proposal $($minimum of two paragraphs$)$ to your company persuading them to purchase your rollercoaster plans. Give your coaster a creative name, catch phrase, and discuss what distinguishes it from other roller coasters in the amusement park. Be creative and persuasive!