Polynomials Graphing and Solving

Standards MM3A1. Students will analyze graphs of polynomial functions of higher degree. a. Graph simple polynomial functions as translations of the function f(x) = ax n. b. Understand the effects of the following on the graph of a polynomial function: degree, lead coefficient, and multiplicity of real zeros. c. Determine whether a polynomial function has symmetry and whether it is even, odd, or neither. d. Investigate and explain characteristics of polynomial functions, including domain and range, intercepts, zeros, relative and absolute extrema, intervals of increase and decrease, and end behavior.

Characteristics A polynomial function is defined as: f(x)= The degree of a polynomial is_______. The coefficient of the variable with the greatest exponent is called the _______.

We will use the “Graphs of Polynomial Functions” handout for the remaining slides

Factors and Intercepts Each of these equations can be re- expressed as a product of linear factors Factor the ones that you can Look at the graphs of the ones that you were able to factor and list the x- intercepts. How are the intercepts related to the linear factors?

Factor this function What are the roots of this fifth order polynomial? Why aren’t there five roots since this is a fifth degree polynomial? Check the roots by graphing on your calculator.

Symmetry Which functions are symmetric about the y-axis? What do you call them? Which functions are symmetric about the origin? What do you call them?

Zeros X-intercepts are called the zeros. Why do you think they have this name? Find the zeros of the functions on the handout. Make a conjecture about the relationship between the number of zeros and the degree of the polynomial. Test your conjecture by graphing y=x 2 (x-1)(x+4) on your calculator. Do you need to change your conjecture?

End Behavior Graph the following on your calculator. Make a rough sketch next to each one.

Answer the following about your functions. Is the degree even or odd? Is the leading coefficient positive or negative? Does the graph rise or fall on the left? Does the graph rise or fall on the right? Write a conjecture about the end behavior (rising or falling on the left and right) of the graph of a function.

Practice Graphing Graph these functions by hand, then check your work on a calculator.

Critical Points - Extrema Use the “Graphs of Polynomial Functions” handout. Determine where the functions are increasing and decreasing. Locate the turning points. Are they relative or absolute extrema? Make a conjecture about the maximum number of turning points and the degree of the polynomial.