Unit 6 A: Polynomial Functions Name: _______________________ Period: ______________________ Section 6.1 Analyzing Graphs of Polynomial Functions Objective(s): Analyze graphs of polynomial functions to determine its characteristics. Essential Question: Explain which term of the polynomial function is most important when determining the end behavior of the function. Homework: Assignment 6.1. #1 – 15 in the homework packet. Notes: The graph of y = x² is a _________________. It’s graph is The graph of y = - x² is Higher Degree Polynomials y = x³ + 4x² - x – 2 or y = 2x4 + 3x³ + 2x² + 3x – 1 are called higher degree polynomials. Notice that their degree (highest power) is more than 2. All of these graphs will be curves with high and/or low points called turning points (also called local maximum or local minimum. The graphs are “snake-like”. Examples are shown below. A B How many turning points are there in C D Graph A?_________Graph B? __________ Graph C? ______________ Graph D?________________ Reflection: 1 Turning Points The graph of a polynomial function has AT MOST n – 1 turning points, where n = the degree (highest power of x). It may have fewer turning points. For the examples, find the maximum number of turning points for each: Example 1: f(x) = x3 + 6x2 + x – 1 _________ Example 2: f(x) = 2x10 – 8x4 + 3x _________ End Behavior for Even Degree Polynomial Functions The graph of y = x² shows end behavior of rising (going up) on both ends. The graph of any parabola y = ax² + bx + c will have the same shape, as long as a is positive. If a is NEGATIVE, the graph of y = ax² + bx + c will have shape. Both ends of the graph are going down (falling). If the degree of the polynomial function is EVEN, it will have the same end behavior as y = ax². Going up (rising) on both ends, or going down (falling) on both ends. There may be turning points in between. They will basically be a “ U ” or “” shape with wiggles (turning points) possibly in the middle. Look at these examples To Review: Degree – even a) Leading coefficient positive Reflection: b) Leading coefficient negative 2 End Behavior for Odd Degree Polynomial Functions If the degree of the polynomial function is ODD and LEADING COEFFICIENT IS POSITIVE it will have the shape shown. The left side is falling and the right side is rising. The sides are going in opposite direction. Remember there may be turning points in between. If the degree of the polynomial function is ODD and LEADING COEFFICIENT IS NEGATIVE (-) it will have the shape shown. The sides are going in opposite direction. Remember there may be turning points in between. To Review: Degree – odd a) Leading coefficient positive b) Leading coefficient negative How many turning points does Graph A have?__________Graph B __________ End Behavior for Polynomial Functions The end behavior is described in mathematical language with symbols like the → arrow, which means “approaches”, and the symbol for infinity ∞. When we write x → + ∞ it means the right side of the graph. When we write x → - ∞ it means the left side of the graph. If a graph is rising (going up), we say f(x) → + ∞ If a graph is falling (going down), we say f(x) → - ∞ The answer to the end behavior of any polynomial function is either + ∞ OR - ∞ Reflection: 3 Example 3: What is the end behavior of f(x) as x → + ∞ (on the right) for the graph shown below? Remember, the answer to end behavior has to be + ∞ or - ∞. f(x) → ______ f(x) → ______ f(x) → ______ Example 4: What is the end behavior of f(x) as x → - ∞ (on the left) for the graphs shown above? (Looking at the left side) f(x) → ______ f(x) → ________ f(x) → _________ Finding the end behavior of f(x) given the polynomial (but no graph). If you are given the polynomial function, you will need to think what the sketch of the graph would look like. Remember that this depends on the degree and if the leading coefficient is positive or negative. (see pages 2 – 3) SKETCH and describe the end behavior of f(x) as x→+ ∞ (i.e. Is the graph going up or down on the right side?) Sketch here Example 5: f(x) = 3x4 – 4x2 + 1 f(x) → _________ Example 6: f(x) = 2x5 + x2 f(x) → _________ Example 7: f(x) = -x7 + 5x4 – x + 2 f(x) → _________ Example 8: f(x) = -x6 + 9x4 + x3 f(x) → _________ Reflection: 4 Describe the end behavior of f(x) as x→- ∞ (i.e. Is the graph going up or down on the left side?) Example 9: f(x) = -2x2 – 4x + 7 f(x) → _________ Example 10: f(x) = -8x3 + 6x2 – 12x – 9 f(x) → _________ Example 11: f(x) = -x5 + 9x4 – 3x2 + 22 f(x) → _________ Example 12: f(x) = 8x4 + x2 + 9 f(x) → _________ Vocabulary Connection: The following statements are equivalent. - Zero: k is a zero of the function f(x). - Factor: (x – k) is a factor of the function f(x). - Solution: k is a solution of the equation f(x) = 0 - x-intercept: if k is a real number, then k is an x-intercept of the graph of the function f(x). Example 13: If 7 is a zero of f(x) = x3 – 7x2 + 2x – 14, then Factor = Solution = x-intercept = ( , 0) Example 14: If -2 is a zero of f(x) = x3 + 3x2 + x – 2, then Factor = Solution = x-intercept = ( , 0) Reflection: 5 Section 6.2 Graphing a Polynomial in Factored Form Objective(s): Graph polynomial functions. Essential Question: What is the greatest number of local maximums and minimums that a cubic function can have? Homework: Assignment 6.2. #16 – 21 in the homework packet. Steps to Graph 1. If the polynomial function is in factored form, place points on the x-axis where the x-intercepts occur. 2. Determine the shape by deciding on the degree. 3. If a factor is raised to an EVEN power, example (x – 3)² the graph will not cross the x-axis at 3. It will TOUCH at 3 on the x-axis and then turn and go the other way. 4. Sketch the graph. Graph the function. Describe the end behavior of f(x). Example 1: f(x) = (x – 2)(x + 3)(x – 6) Describe the end behavior of f(x) as x→+ ∞ f(x) → _________ Describe the end behavior of f(x) as f(x) → _________ Reflection: Example 2: f(x) = x(x + 2)(x + 4)(x + 6) Describe the end behavior of f(x) as x→+ ∞ f(x) → _________ x→- ∞ Describe the end behavior of f(x) as x→- ∞ f(x) → _________ 6 Example 3: f(x) = -(2x + 1)(x – 4)(x – 5)(x + 3) Describe the end behavior of f(x) as x→+ ∞ f(x) → _________ Describe the end behavior of f(x) as Example 4: f(x) = -(x + 4)(x – 2)(2x + 3) Describe the end behavior of f(x) as x→+ ∞ f(x) → _________ x→- ∞ Describe the end behavior of f(x) as x→- ∞ f(x) → _________ f(x) → _________ Example 5: f(x) = (x – 1)² (x + 3) (x+4) Example 6: f(x) = x(x – 1)² (x – 4)(x +2) Describe the end behavior of f(x) as Describe the end behavior of f(x) as x→+ ∞ f(x) → _________ Describe the end behavior of f(x) as f(x) → _________ Reflection: x→+ ∞ f(x) → _________ x→- ∞ Describe the end behavior of f(x) as x→- ∞ f(x) → _________ 7 Sample CCSD Common Exam Practice Question(s): 1. Which best represents the graph of the polynomial function y x 4 2 x 2 2 x 3 ? 2. Which describes the end behavior of the graph of f x x 4 5 x 2 as x ? A. f x B. f x C. f x 0 D. f x 2 3. Use the graph of a polynomial function below. What are the zeros of the polynomial? Reflection: A. {2} B. {2} C. {3, 1, 4} D. {3, 1, 4} 8 Section 6.3 Using Properties of Exponents Objective(s): Use properties of exponents to simplify and evaluate expressions. Essential Question: Which properties of exponents require you to check that two or more bases are the same before applying the property? Homework: Assignment 6.3. #22 – 37 in the homework packet. Notes: How many factors of 2 are there in the product 23 25 ? Use your answer to write the product as a single power of 2. Write each product as a single power. 22 25 = 33 36 = 71 7 6 = x4 x4 = Write each quotient as a single power of 2 by first writing the numerator and denominator in “expanded form” (for example, 23 2 2 2 ) and then canceling common factors. 23 = 21 26 = 22 23 = 27 Properties of Exponents a m a n a mn (ab)m a mbm am a mn n a am a bm b (a m )n a mn a0 1 am m 1 am Reflection: 9 Simplify. Write the answer with positive exponents. Example 1: 7 3 4 Example 2: 3x y 2 4 3 Example 3: y6 y8 Example 4: 4x 0 Example 5: 43 Example 6: 6x 4 Example 7: r 5 s 2 Example 8: 3x 2 y 4 z 8 6 xy 6 z 2 Example 9: 6 x15 y12 z 4 9 x 7 y 2 z 4 Reflection: 10 Section 6.4 Adding, Subtracting, and Multiplying Polynomials Objective(s): Simplify polynomial expressions. Essential Question: What is the advantage of the box method when multiplying polynomials? Homework: Assignment 6.4. #38 – 52 in the homework packet. Notes: Perform the indicated operation. Example 1: (10x3 – 5x + 8) + (-6x2 + 12x + 3) Example 2: 6x – (14 – 4x) Example 3: (-4x2 + 5) – (-x3 – 10x2 – 4) Multiply Example 4: -10x(6x – 8) Example 5: (3x – 10)(x – 6) Example 6: (2x – 3y)2 Example 7: (x + 8y)2 Example 8: (6x – 4)(2x2 + 7x – 9) Reflection: 11 Example 9: (2x + 4)(x2 – 3x – 1) Example 10: (2x – 3)(x – 7)(x + 5) Sample CCSD Common Exam Practice Question(s): 1. Which polynomial represents the product of x 2 x 3 8 ? A. x 4 2 x3 8 x 16 B. x 4 2 x3 8 x 16 C. x 4 16 D. x 4 16 Reflection: 12 Section 6.5 Factoring and Solving Polynomial Equations Objective(s): Solve polynomial equations by factoring. Essential Question: Give an example of a binomial that can be factored either as the difference of two squares or as the difference of two cubes. Show the complete factorization of your binomial. Homework: Assignment 6.5. #53 – 70 in the homework packet. Notes: General Steps to Factor 1. Factor out the GCF first (if possible) 2. Binomial: Difference of squares a2 – b2 = (a + b)(a – b) 3. Binomial: Sum/Difference of cubes a3 + b3 = (a + b)(a2 – ab + b2) a3 – b3 = (a – b)(a2 + ab + b2) 4. Trinomial: Use ac method, box method, etc. 5. Four terms: Factor by grouping Factor the polynomial completely. Example 1: 18x3 – 6x2y + 10xz3 Example 2: 3x + 21 + xy + 7y Example 3: x2 – 9x + 14 Example 4: x2 – 3x – 88 Example 5: x2 + x – 20 Example 6: x2 – 49 Reflection: 13 Example 7: 9x2 – 25y2 Example 8: 3x2 – 75 Example 9: 7x2 – 5x – 2 Example 10: 6x2 – 26x – 20 Example 11: x4 – 7x2 + 6 Example 12: x3 + 27 Example 13: 8x3 – 125 Example 14: 27x3 + 1 Example 15: 2y3 + 54z3 Reflection: 14 Solve the equation. Example 16: 2(x – 4)(x + 5) = 0 Example 17: x2 + 3x - 40 = 0 Example 18: x2 + 7x = 60 Example 19: 2x3 + 18x2 + 28x = 0 Example 20: x3 – x = -8x2 + 8 Example 21: x3 + 5x2 = 50x Example 22: x4 – 18x2 + 32 = 0 Example 23: x4 – 5x2 – 36 = 0 Reflection: 15 Sample CCSD Common Exam Practice Question(s): 1. Which of the following represents the solution set of the polynomial equation x 4 7 x 2 12 0 ? A. B. C. D. 2, 2, i 3, i 3 2, 2, 3, 3 2i, 2i, i 3, i 3 2i, 2i, 3, 3 2. What is the factored form of the polynomial x 3 27 ? A. x 3 x 2 3 x 9 B. x 3 x 2 3 x 9 C. x 3 x 2 3 x 9 D. x 3 x 2 3 x 9 Reflection: 16