MODELING WITH POLYNOMIAL FUNCTIONS: LESSON 26 The purpose of this lesson is to learn new possibilities for table and graph patterns, equations, inequalities, and algebraic operations that arise from consideration of polynomials of degree greater than two.

LESSON OBJECTIVES: Model problem situations using polynomial functions. Identify patterns relating rules and graphs of polynomial functions-connecting polynomial degree to local maximum and local minimum values and zeroes.

Polynomial and Rational Functions: Think About This Situation Worksheet Look at the two graphs and answer the questions on the worksheet.

Both sketches have pieces that look somewhat like parabolas, so one could expect that quadratic functions might be pieced together to provide useful models. Also linear, exponential or inverse power models.

A set of axes could be drawn on the sketches and a scale could be set A set of axes could be drawn on the sketches and a scale could be set. Then several points on each could be identified with coordinates and a calculator or spreadsheet regression tool could be used to find the function to model the sketch.

The points on each graph where the graph changes direction (local maximum or local minimum points) and several points in between are more critical than those on the far left or far right of each sketch.

A polynomial function is a function that can be defined by evaluating a polynomial. A function ƒ of one argument is called a polynomial function if it satisfies the equation for all arguments x, where n is a non-negative integer and a0, a1,a2, etc, are constant coefficients.

What kinds of graphs do functions have? Linear Function Quadratic Function

Exponential Function Rational Function

Logarithmic Function Polynomial Function

Polynomial expression: consisting of a sum of terms each of which is the product of a constant and one or more variables raised to a positive or zero integral power. For one variable, x, the general form is given by: a0xn + a1xn-1 + … + an-1 x + an, where a0, a1, etc., are real numbers

Look at the Section I Design and answer the following questions. What key points on the graph do you think would be helpful in finding a cubic function that models the proposed Section I Design? Compare the graph of the cubic function to the shape of the Section I Design.

What do you think might be the minimum number of points needed to find a cubic model for a data or graph pattern?

As you may have noted, there are some significant differences between the design ideas for Section I and II of the proposed roller coaster. Find coordinates of key points outlining the shape of this graph.

Then find the cubic function model for the pattern in those points and compare its graph to the shape of the proposed Section II graph. What do you think might be the minimum number of points needed to find a quartic model for a data or graph pattern?

Find a quartic function model for the pattern of data points and check how well its graph matches with the pattern in the Section II Design.

Local Maximum: Peak A function f(x) has a local maximum at x0 if and only if there exists some interval containing x0 such that f(a) ≥ f(x) for all x in the interval. The height of the function at "a" is greater than (or equal to) the height anywhere else in that interval.

Local Minimum: Valley A function f(x) has a local minimum at x0 if and only if there exists some interval containing x0 such that f(a) ≤ f(x) for all x in the interval. The height of the function at "a" is less than (or equal to) the height anywhere else in that interval.

It may not be the minimum or maximum for the whole function, but locally it is.

Degree of a Polynomial: The highest degree of any term in the polynomial. Examples: POLYNOMIAL DEGREE x2 – 2x + 3 2 x7 + x + 4 7 x2 – y4 4 ab + 9 5x + 2 1 3

Identify the degree of each function. 𝑔 𝑥 =2𝑥3 + 8𝑥2 +3𝑥 - 2 h 𝑥 =𝑥4 - 8𝑥3+16𝑥2 + 4 Estimate coordinates of the local maximum and local minimum points on graphs of the functions.

Why are the points (0, 0) and (4 Why are the points (0, 0) and (4.25, 7) not considered local minimum or local maximum for the cubic function in Section I Design for the roller coaster? Why are the points (0, 25) and (5, 10) not considered local minimum or local maximum for the quartic function in Section II Design for the roller coaster?

Homework “Summarize the Mathematics” worksheet. “Polynomial and Rational Functions” worksheet.