2, or x 5, 3 x 0, x 2

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1 Pre-AP Algebra 2 Lesson 2 End Behavior and Polynomial Inequalities Objectives: Students will be able to: use a number line model to sketch polynomials that have repeated roots. use a number line model to solve factored form polynomial inequalities and write the solution in interval and inequality notation. describe the end behavior of polynomial function using as x, f(x) notation. determine the end behavior of a non-factored polynomial function by looking at the sign of the leading coefficient and whether the degree is odd or even. Materials: Hw #7-1 answers overhead; tally sheets; note-taking templates; pair work packet; homework #7-2 Time Activity 5 min Homework Review Show answer to hw #7-1 on the overhead. Pass around tally sheets. 10 min Homework Presentation Review top 2 or 3 problems from tally sheets. Discuss the last problem, and what effect a repeated root can have on the shape of a polynomial function s graph. 25 min Direct Instruction Lesson 2: End Behavior & Solving Polynomial Inequalities Section: Polynomials and Exponents Example: Create a number line model for y = (2 x)(x + 5)(x)(x + 3). Concepts: 1) End behavior: what does the function do as x gets very positive/negative? With a polynomial, the function will go up to infinity, or down to negative infinity. Use this notation: as x, f(x) Example: For the given example, the end behavior is: x +, f(x) - x -, f(x) - Concepts: 2) Inequalities: Make a number line model Use the number line to write the solution. Pay attention to parentheses/brackets. Example: Solve (2 x)(x + 5)(x)(x + 3) < 0. Based on the model made in example 1, the solution is:, 5 U 3,0 U 2, or x 5, 3 x 0, x 2 35 min Pair Work Hand out the Polynomials Practice worksheet. Students should work individually or in pairs. The last section of the packet is a mini-exploration to see how end behavior works when the polynomial is not factored. 5 min Review Exploration On the overhead, summarize what they should have seen in the exploration (odd/even degree, positive/negative leading coefficient). As x, what happens to the sign of the leading term? That tells you what the end behavior will be. Homework #7-2: Prepare for Quiz 1

2 Pre-AP Algebra 2 Pair Work Name: Polynomials Practice 1) Make a number line model for each polynomial function. Sketch a reasonable graph on the number line. Check your sketch by graphing the function on the calculator. a. f (x) (x 2)(x 5) 2 b. g(x) (2x 1)(3x 8) c. h(x) (x 1)(x 2)(x 3)(x 1) 2 2) Use you work in problem 1 to solve these inequalities. You don t need to do any more calculations. a. (x 2)(x 5) 2 0 b. (2x 1)(3x 8) 0 c. (x 1)(x 2)(x 3)(x 1) 2 0 d. (x 1)(x 2)(x 3)(x 1) 2 0

3 3) Look back at your work in problem 1 to describe the end behavior of each function. f (x) (x 2)(x 5) 2 As x +, f(x) As x -, f(x) g(x) (2x 1)(3x 8) h(x) (x 1)(x 2)(x 3)(x 1) 2 Exploration: How can you determine the end behavior of a polynomial function in standard form? 1) Given the function f (x) x 3, for what values of x will f(x) be positive? Negative? 2) Given the function f (x) x 4, for what values of x will f(x) be positive? Negative? 3) If f (x) x 3, as x +, f(x), and as x -, f(x). You can do this by thinking about your answers to question 1 (imagine plugging in a really big positive or negative number into x). 4) If f (x) x 4, as x +, f(x), and as x -, f(x). You can do this by thinking about your answers to question 2. 5) Will f (x) x 32 have end behavior like f (x) x 3 or f (x) x 4? Why? 6) Will f (x) x 47 have end behavior like f (x) x 3 or f (x) x 4? Why? 7) How would putting a negative sign in front of the leading coefficient change the graph? How would that affect the end behavior? 8) In a polynomial function, only the leading term matters when trying to determine the end behavior. Using this fact, and your answers to questions 1 7, predict the end behavior of each of these functions. Then, graph them on the calculator to see how you did. a. f (x) 4x 7 3x 5 1 As x +, f(x) As x -, f(x) b. f (x) 2x 10 x 3 As x +, f(x) As x -, f(x) c. f (x) 1 2 x9 As x +, f(x) As x -, f(x) d. f (x) 0.2x 22 3x 5 2x As x +, f(x) As x -, f(x)

4 Pre-AP Algebra 2 Homework #7-2 Name: Hw #7-2: Polynomials Review QUIZ 1 Part 1: For each expression, determine if it is a polynomial or not. If it is, write it in standard form. Then, describe it by degree and number of terms, and identify the leading coefficient. a) 4x 2 3 x2 b) 3x 2 5x c) 23x 3 d) 2x 5x 3 10x 4 Part 2: Make a number line model for each polynomial and sketch a reasonable graph of the function p(x). a) p(x) (2x 7)(x 5)(x 6) b) p(x) (x 4)(x 3) 2 (1 x) c) p(x) f (x)g(x)h(x)

5 Part 3: Describe the end behavior of each function. Look at your work in part 2 to help with the first two. Look at your pair work from class to help you with the second two. a) f (x) (2x 7)(x 5)(x 6) b) f (x) (x 4)(x 3) 2 (1 x) c) f (x) 4x 21 3x 15 5x d) f (x) 2x 18 5x 9 Part 4: Solve each inequality by making a number line model. Write your solutions in both interval and inequality notations. a) (3 x)(x 5) 0 b) (x 1)(x 2) 2 (x 3) 0 c) 6x 2 7x 5 0 (You ll need to factor the left side to solve this inequality.)

6 HW #7-1 Tally Sheet Part 1: 1) 2) 3) 4) Part 2: 1) 2) 3) 4) 5) Part 3: 1) 2) 3) 4) 5)

7 HW #7-1 Answer Sheet Part 1: ) 3 4 8x 5x, cubic binomial 2) x 2x 5, quartic trinomial 3 2 3) 2x, quadratic monomial 4) x 3 10x 2 4x 40, cubic 4-term polyn. Part 2: 1) -2, 1, 3 2) (, 2),( 2,1),(1,3)(3, ) 3) Part 3: 1) 2) 3) 4) 5)