Lesson 5: 4-5 Analyzing Graphs of Polynomial Functions

1. Why do you need so many more data points to graph higher order equations than you do for linear or quadratic functions?
2. How do you find out where the function might cross the x-axis?
3. What is the difference between a relative maximum and an extreme maximum?
4. What is a turning point?

Location Principle

F.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.

F.IF.7.c Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

SAT questions related to polynomial operations: 7-3-2, 6-4-1, 4-3-5, 3-4-6, 1-3-5, 4-4-2, 3-4-33, 8-3-5; dividing polynomials: 7-3-13, 2-3-15

Tables of values can be used to explore two types of changes in the values of a polynomial function. A change of signs in the value of f(x) from one value of x to the next indicates that the graph of the function crosses the x-axis between the two x-values. A change between increasing values and decreasing values indicates that the graph is turning for that interval. A turning point on a graph is a relative maximum or minimum.

Annual attendance at the movies has fluctuated since the first movie theater opened in 1906. Overall movie attendance peaked during the 1920s, and it was at its lowest during the 1970s. A graph of the annual attendance to the movies can be represented by a polynomial function.

Recall that the degree of he function is also the maximum number of zeros the function can have.

Some odd functions, like f(x)=x^3, have no turning points.

A polynomial with a degree greater than 3 may have more than one relative maximum or relative minimum.

Zeros and turning points will not always occur at integral values of x.

Have students discuss the appropriateness of describing real-world situations with mathematical functions. Help them to understand that a function is usually just an approximation of the real-world data, and is often only a reasonable model of a limited domain of values.

McGraw Hill resources

McGraw Hill resources