Task $1$: A new dive roller coaster is to be created, called the $"4$U Striker". Sketch a graph that could model the path of the new $"4$U Striker" rollercoaster over its first $20$ seconds in motion. Let x represent time $($seconds$)$ since the ride begins $($at the launching ramp$).$ Let y represent the height of the roller coaster ride in relation to the launching ramp $($feet$).$

Your model must include the following:

The launching ramp is $40$ft above the ground

The coaster must climb at least twice in a $20$ second interval.

As its first thrill, the coaster will drop down from the launching ramp.

It reaches a height of $150$ft above the launching ramp after $6$ seconds A local maximum at a height of $180$ft above the launching ramp

Contains at least $2$ positive intervals and $2$ negative intervals

The coaster does not always reach the same height with each climb and drop. Task $2$: Determining the equation of a function to model the path of the rollercoaster.

What type of function could model the coaster's height above the launching ramp? $($circle one$)$

Rational

Quadratic

Logarithmic

Sinusoidal

Polynomial

Give reasons for your choice?

$2.$ What information do you need in order to write the equation of a function that could model the coaster's path?$3.$ Determine the equation of your model that will pass through the point $(6,150.)$ Show all your work.Task $3$: Another coaster at the amusement park follows a path given by the equation listed below.$h(x)=-\frac{1}{3}{x}^4+\frac{46}{9}{x}^3-\frac{208}{9}{x}^2+\frac{118}{3}x+29$ During what times$),$ within the first $10$ seconds of the ride, will the coaster be above a height of $50$ feet from its launching ramp?