Polynomial Functions and Models 4.2 Understand the graphs of polynomial functions. Evaluate and graph piecewise-defined functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Graphs of Polynomial Functions Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Constant Polynomial Function Has no x-intercepts or turning points Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Linear Polynomial Function Degree 1 and one x-intercept and no turning points. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Quadratic Polynomial Functions Degree 2, parabola that opens up or down. Can have zero, one or two x-intercepts. Has exactly one turning point, which is also the vertex. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Cubic Polynomial Functions Degree 3, can have zero or two turning points. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Quartic Polynomial Functions Degree 4, can have up to four x-intercepts and three turning points. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Quintic Polynomial Functions Degree 5, may have up to five x-intercepts and four turning points. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Degree, x-intercepts, and turning points The graph of a polynomial function of degree n  1 has at most n x-intercepts and at most n  1 turning points. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Use the graph of the polynomial function shown. a) How many turning points and x-intercepts are there? b) Is the leading coefficient a positive or negative? Is the degree odd or even? c) Determine the minimum degree of f. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Group Work Graph f(x) = 2x3  5x2  5x + 7, and then complete the following. a) Identify the x-intercepts. b) Approximate the coordinates of any turning points to the nearest hundredth. c) Use the turning points to approximate any local extrema. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Let f(x) = 3x4 + 5x3  2x2. a) Give the degree and leading coefficient. b) State the end behavior of the graph of f. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Piecewise-Defined Polynomial Functions Example Evaluate f(x) at 6, 0, and 4. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example Complete the following. a) Sketch the graph of f. b) Determine if f is continuous on its domain. c) Evaluate f(1). Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Copyright © 2006 Pearson Education, Inc Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley