Advanced Algebra
Chapter 8, Polynomials and Polynomial Functions
The graph of a polynomial is a curve. Designers may be able to model the shapes of their designs with polynomials, then use the models to analyze and perhaps improve what they have made.
Part A, Graphing Polynomial Functions;
Part B, Maximums and Minimums
1. Look at the photo of the roller coaster known as Python.
a. Explain why the shape of the portion of Python shown in the photo cannot be modeled by a polynomial function.
b. Any function that models the curve of a roller coaster track must be continuous. Why?
c. The graph below models the start of the roller coaster Desperado. Find Desperado's height and longest drop. Then identify the relative maximum and the relative minimum in the interval shown on the graph.
d. Graph the "roller coaster" curve y = 0.75 x 4x 3 17.25 x2 + 15.67 x. TRACE to find the absolute maximum, the relative maximum, and the relative minimum.
Part C, Zeros of a Function;
Part D, Making Connections
2. The Indianapolis 500 auto race track consists of four straightaways separated by quarter-circles at the corners.
a. Find the lengths of the straightaways, the semicircles, and the entire track. Draw a picture that shows the track and its dimensions.
b. Find the measurements and area of the rectangular "infield" (shaded area) of the track.
c. A track designer has been hired to see if the area of the infield can be increased to make room for more spectators. The idea is to change the lengths of the long straightaways and to convert the end sections to semicircles. The length of the track must stay the same. Express the length of the track in terms of x and r. (straightaway = x; radius = r)
d. Equate the above expression to the actual length of the track and solve for r.
e. Express the area of the infield A as a polynomial in x.
f. Graph the polynomial.
g. Find the measurements and area of the redesigned infield with maximum area.
h. Should the new track design be adopted?
