1 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. START UP Day Algebraically determine the Zeros of each function: (Hint: FACTOR) 2. Can you describe each graph?

2 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Objective: Students will be able to solve problems involving Power and Polynomial functions. Essential Questions: What is a Power Function? What are the general characteristics of the graphs of Polynomial Functions? What is the leading coefficient test? Home Learning: p. 182 # p. 193 #2, 3, 24, 28, 49, 51, 69-74

3 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. 2.2 & 23 Power Functions & Polynomial Functions of Higher Degree with Modeling Demana, Waits, Foley, Kennedy

4 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Power Function Any function that can be written in the form f(x) = k x a, where k and a are nonzero constants, is a power function. The constant a is the power, and k is the constant of variation, or constant of proportion. We say f(x) varies as the a th power of x, or f(x) is proportional to the a th power of x.

5 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Analyzing Power Functions

6 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Solution

7 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Monomial Function Any function that can be written as f(x) = k or f(x) = k·x n, where k is a constant and n is a positive integer, is a monomial function.

8 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Example: Graphing Monomial Functions

9 Copyright © 2015, 2011, and 2007 Pearson Education, Inc.

10 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Your Turn:

11 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. End Behavior Math Lab--Partner Activity 15 Minutes! Use a Graphing Calculator to produce your graphs. Sketch each Look for connections & discuss what you see to complete each statement of conclusion.

12 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Taking it to the Limit— Which way will it go? Leading Coefficient Test DEGREE OF POLYNOMIAL (Highest Exponent) ”n” +a-a Even DEGREE Odd DEGREE

13 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. LOCAL EXTREMA—Local minimums and/or maximums OR “stationary points” At most, a graph of a polynomial function to the “n” power, has “n-1” local extrema ZEROS, ROOTS or X-INTERCEPTS: At most, a graph of a polynomial function to the “n” power, has “n” real zeros. An even power function may not have any real zeros An odd power function always has at least one real zero

14 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. In Search of the Real Zeros Sometimes your polynomial will be factorable! Consider setting your polynomial = 0 Factor your polynomial completely Use the ZERO PRODUCT PROPERTY and solve for all the possible “x” values These will be your x- intercepts or your zeros

15 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Graphing: 1. Plot your “zeros” 2. Use your knowledge of “end behavior”. 3. Connect with a smooth curve. Zeros with multiplicity of 2? These will be a point of tangency— They don’t pass through! Zeros with multiplicity of 3? These will be a point of inflection.

16 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. Graphing: 1. Plot your “zeros” 2. Use your knowledge of “end behavior”. 3. Connect with a smooth curve.

17 Copyright © 2015, 2011, and 2007 Pearson Education, Inc. YOUR Turn: End Behavior? Zeros?How many and identify all Relative Extrema? How many might there be? Sketch it

18 Copyright © 2015, 2011, and 2007 Pearson Education, Inc.