2.2 Polynomial Functions 2015/16 Digital Lesson

HWQ 8/17 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 2

Polynomial Functions are Continuous and Smooth y x –2 2 y x 2 y x 2 Functions that are not continuous are not polynomial functions (Piecewise) Functions that have sharp turns are not polynomial functions (Absolute Value) Polynomial functions have graphs that are continuous and smooth

The polynomial functions that have the simplest graphs are monomials of the form If n is even-the graph is similar to If n is odd-the graph is similar to For n-odd, the greater the value of n, the flatter the graph near(0,0) y x –2 2

Transformations of Monomial Functions Example 1: The degree is odd, the negative coefficient reflects the graph on the x-axis, this graph is similar to

Transformations of Monomial Functions Example 2: The degree is even, and has as upward shift of one unit of the graph of

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Polynomial Function A polynomial function is a function of the form where n is a nonnegative integer and each a i (i = 0, , n) is a real number. The polynomial function has a leading coefficient a n and degree n. Examples: Find the leading coefficient and degree of each polynomial function. Polynomial FunctionLeading Coefficient Degree The Leading Coefficient Test (End Behavior Test)

The graph of a polynomial eventually rises or falls. This can be determined by the function’s degree (odd or even) and by its leading coefficient (positive or negative) y x –2 2 When the degree is odd: If the leading coefficient is positive The graph falls to the left and rises to the right If the leading coefficient is negative The graph rises to the left and falls to the right

y x –2 2 When the degree is even: If the leading coefficient is positive The graph rises to the left and rises to the right If the leading coefficient is negative The graph falls to the left and falls to the right

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Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 11 Zeros of a Function A real number a is a zero of a function y = f (x) if and only if f (a) = 0. A polynomial function of degree n has at most n real zeros. Real Zeros of Polynomial Functions If y = f (x) is a polynomial function and a is a real number then the following statements are equivalent. 1. x = a is a zero of f. 2. x = a is a solution of the polynomial equation f (x) = (x – a) is a factor of the polynomial f (x). 4. (a, 0) is an x-intercept of the graph of y = f (x).

Zeros of Polynomial Functions The graph of f has at most n-1 relative extrema (relative minima or maxima.) Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 12 A polynomial function of degree n has at most n real zeros. It has exactly n total zeros (both real and imaginary.)

Use the leading coefficient test (end behavior test) to describe left and right hand behavior and sketch the graph Example 1 y x –2 2

Use the leading coefficient test (end behavior test) to describe left and right hand behavior and sketch the graph Example 2 y x –2 2

Use the leading coefficient test (end behavior test) to describe left and right hand behavior and sketch the graph Example 3 y x –2 2

Apply What You Know! Given: End Behavior Test Right Side____ Left Side_____ Does it shift?_____ Draw the graph!

Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 17 y x –2 2 Example: Real Zeros Example: Find all the real zeros of f (x) = x 4 – x 3 – 2x 2. Factor completely: f (x) = x 4 – x 3 – 2x 2 = x 2 (x + 1)(x – 2). The real zeros are x = –1, x = 0, and x = 2. Notice that there is a zero at x = 0 that has a multiplicity of 2. A zero with an even multiplicity will bounce off the x-axis. f (x) = x 4 – x 3 – 2x 2 (–1, 0) (0, 0) (2, 0)

Finding Zeros of a Polynomial Function Student Example Find all real zeros of Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 18

Example continued: Sketching the graph of a Polynomial Function with known zeros: Sketch a graph by hand. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 19 y x –2 2

Analyzing a Polynomial Function Find all real zeros and relative extrema of: Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 20

Mulitiplicity of Zeros How many zeros should each polynomial have? What conclusion do you reach when you graph them?

Finding a Polynomial Function with Given Zeros Write an equation for a polynomial function with zeros at x = -2, 1, and 3. Sketch a graph by hand. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 22 y x –2 2

Finding a Polynomial Function with Given Zeros Student Example: Find a polynomial function with the given zeros: x = -1, 2, 2 Sketch a graph by hand. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 23 y x –2 2

Sketching the Graph of a Polynomial Function Sketch the graph of 1.What is the end behavior? 2.Find the zeros of the polynomial function. 3.Plot a few additional points. 4.Draw the graph. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 24 y x –2 2

Sketching the Graph of a Polynomial Function Sketch the graph of 1.What is the end behavior? 2.Find the zeros. 3.Plot a few additional points. 4.Draw the graph. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 25 y x –2 2

Homework Section 2.2 pg odd, odd, odd, 61 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 26