• Recognize polynomial functions, noting their degree and leading coefficient
• Apply various methods to divide polynomials and locate the zeros of polynomial equations
• Generate graphs and formulate equations for polynomial functions
• Explain the end behavior of polynomial functions

Ride the Polynomial: Roller Coaster Curves and Calculations

Jordan, a civil engineer, is designing a new roller coaster. The track’s shape, characterized by its exhilarating drops and climbs, can be modeled using polynomial functions. The design must prioritize safety while maximizing excitement and adhering to material constraints.

 

As Jordan begins the design, the initial drop is the first consideration, setting the tone for the entire ride. He sketches the initial drop modeled by the polynomial function [latex]f(x)=−0.5x^4+3x^3+x^2[/latex]. The function not only dictates the steepness and curvature of this descent but also encapsulates the delicate balance between thrill and safety. This leads us to our first task: dissecting this function to understand how its degree and leading coefficient will shape the riders’ experience.

Having explored the initial drop’s polynomial and its impact on the thrill factor, we now ascend to the first hill of our roller coaster. This is where we experience the anticipation build-up, a crucial element of the ride’s excitement. As we crest the hill, let’s shift our focus to the structural integrity and the importance of zeros in the polynomial function that models this ascent.

The first hill following the drop can be modeled by the polynomial function [latex]g(x)=x^3−4x^2+4x[/latex].

With the zeros of the first hill pinpointed, ensuring a safe and exhilarating peak, we now dip into the smaller bumps that add variety to our ride. These minor undulations are essential for maintaining rider engagement before the next major feature. Let’s calculate the vertex of the polynomial function representing this smaller bump, which will tell us the maximum height riders will experience, ensuring it provides excitement while keeping the ride’s momentum.

A smaller bump on the track is represented by the polynomial function [latex]h(x)=2x^2−8x+6[/latex].