Analysis POLYNOMIAL PS Fall2015

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1 Analysis POLYNOMIAL PS Fall2015 NAME: SCORE: INSTRUCTIONS PLEASE RESPOND THOUGHFULLY TO ALL OF THE PROMPTS IN THIS PACKET. TO COMPLETE THE POLYNOMIAL PROBLEM SET, YOU WILL NEED TO: 1. FILL IN THE KNOWLEDGE & COMPREHENSION COMPENDIUM. 2. WORK OUT APPLICATION & ANALYSIS PRACTICE PROBLEMS FROM EUREKA MATH LESSON 14 PROBLEM SET #1-7 AND LESSON 15 PROBLEM SET # WORK OUT SYNTHESIS & EVALUATION PRACTICE PROBLEMS #1-2. FOR EXTRA PRACTICE IN PREPARATION FOR MAJOR ASSESSMENTS, PLEASE CONSIDER DOING ANY OF THE FOLLOWING: 1. KC READINESS ASSESSMENT: WRITING EACH KC COMPENDIUM PROMPT ON ONE SIDE OF A FLASH CARD AND YOUR RESPONSE ON THE OTHER SIDE. THEN, QUIZ YOURSELF NIGHTLY BEFORE GOING TO BED UNTIL YOU RECALL THEM ALL PERFECTLY. 2. AA READINESS ASSESSMENT: WORK OUT SOLUTIONS TO THE ODD- NUMBERED EXERCISES IN SECTION 2.5 OF YOUR PRECALCULUS TEXTBOOK, AND CHECK YOUR ANSWERS IN THE APPENDIX. AND/OR SIGN UP FOR A FREE ACCOUNT ONLINE AT KHANACADEMY.ORG. CHOOSE THE WORLD OF MATH MISSION, AND ACHIEVE PRACTICED LEVEL 1 ON THE FOLLOWING EXERCISES: 3. SE ASSESSMENT: REDO THE SE PRACTICE PROBLEMS #1-2 MAKING DIFFERENT CHOICES THAN YOU MADE THE FIRST TIME.

2 Analysis Polynomial PS: KC Compendium Fa2015 Directions: Engage your Knowledge and Comprehension (KC) learning during the introductory exploration on Polynomial Analysis by actively listening for and committing here to a written record of the information that accurately responds to the following prompts. Memorize this information by reviewing it for homework every evening before the KC Readiness Assessment. Polynomial Functions 1. Give an example of a degree-3 polynomial function, and label its lead and constant terms. 2. Paraphrase the Linear Factorization Theorem. 3. Paraphrase the Rational Zeros Theorem. 4. Paraphrase the Remainder Theorem. 5. Give an example of complex conjugates, and paraphrase the Complex Zeros Theorem. 6. Illustrate the five polynomial end behaviors, and label the polynomial properties of each. 7. State the conditions causing the graph of polynomial function to behave like its end behavior. 8. State the two distinct conditions causing a polynomial function to pass through or bounce on the x-axis. 9. State the condition causing a y-intercept in a polynomial function. 10. Recall the graphical and analytical properties of even functions.

3 Lesson 14 Lesson Summary A polynomial of degree may have up to -intercepts and up to 1 relative maximum/minimum points. The function has a relative maximum at if there is an open interval around so that for all in that interval, ( ) ( ). That is, looking near the point, ( ) on the graph of, there is no point higher than, ( ) in that region. The value ( ) is a relative maximum value. The function has a relative minimum at if there is an open interval around so that for all in that interval, ( ) ( ). That is, looking near the point, ( ) on the graph of, there is no point lower than, ( ) in that region. The value ( ) is a relative minimum value. The plural of maximum is maxima, and the plural of minimum is minima. Problem Set 1. For each function below, identify the largest possible number of -intercepts and the largest possible number of relative maxima and minima based on the degree of the polynomial. Then use a calculator or graphing utility to graph the function and find the actual number of -intercepts and relative maxima and minima. a. ( ) = b. ( ) = 4 +4 c. ( ) = Function Largest number of -intercepts Largest number of relative max/min Actual number of -intercepts Actual number of relative max/min a. b. c. Lesson 14: Graphing Factored Polynomials S.76

4 Lesson Sketch a graph of the function ( ) = 1 ( + 5)( + 1)( 2) by finding the zeros and determining the sign of the 2 values of the function between zeros. 3. Sketch a graph of the function ( ) = ( +2)( 4)( 6) by finding the zeros and determining the sign of the values of the function between zeros. Lesson 14: Graphing Factored Polynomials S.77

5 Lesson Sketch a graph of the function ( ) = 2 +2 by finding the zeros and determining the sign of the values of the function between zeros. 5. Sketch a graph of the function ( ) = by determining the sign of the values of the function between the zeros 1, 1, and 3. Lesson 14: Graphing Factored Polynomials S.78

6 Lesson A function has zeros at 1, 3, and 5. We know that ( 2) and (2) are negative, while (4) and (6) are positive. Sketch a graph of. 7. The function ( ) = 16t + 33t + 45 represents the height of a ball tossed upward from the roof of a building 45 feet in the air after seconds. Without graphing, determine when the ball will hit the ground. Lesson 14: Graphing Factored Polynomials S.79

7 Lesson 15 Problem Set 1. Graph the functions from the Opening Exercise simultaneously using a graphing utility and zoom in at the origin. a. At = 0.5, order the values of the functions from least to greatest. b. At = 2.5, order the values of the functions from least to greatest. c. Identify the -value(s) where the order reverses. Write a brief sentence on why you think this switch occurs. 2. The National Agricultural Statistics Service (NASS) is an agency within the USDA that collects and analyzes data covering virtually every aspect of agriculture in the United States. The following table contains information on the amount (in tons) of the following vegetables produced in the U.S. from for processing into canned, frozen, and packaged foods: lima beans, snap beans, beets, cabbage, sweet corn, cucumbers, green peas, spinach, and tomatoes. Year a. Plot the data using a graphing utility. Vegetable Production by Year Vegetable Production (tons) ,393, ,450, ,444, ,151, ,236, ,904, ,313,150 Source: NASS Statistics of Vegetables and Melons, 1995, Table Statistics/ /agr95 4.pdf b. Determine if the data display the characteristics of an odd- or even-degree polynomial function. c. List two possible reasons the data might have such a shape. Lesson 15: Structure in Graphs of Polynomial Functions S.84

8 Lesson The U.S. Energy Information Administration (EIA) is responsible for collecting and analyzing information about energy production and use in the United States and for informing policy makers and the public about issues of energy, the economy, and the environment. The following table contains data from the EIA about natural gas consumption from , measured in millions of cubic feet. U.S. Natural Gas Consumption by Year Year U.S. natural gas total consumption (millions of cubic feet) Source: U.S. Energy Information Administration. a. Plot the data using a graphing utility. b. Determine if the data display the characteristics of an odd- or even-degree polynomial function. c. List two possible reasons the data might have such a shape. 4. We use the term even function when a function satisfies the equation ( ) = ( ) for every number in its domain. Consider the function ( ) = Note that the degree of the function is even, and each term is of an even degree (the constant term is degree 0. a. Graph the function using a graphing utility. b. Does this graph display any symmetry? c. Evaluate ( ). d. Is an even function? Explain how you know. 5. We use the term odd function when a function satisfies the equation ( ) = ( ) for every number in its domain. Consider the function ( ) =3 4. The degree of the function is odd, and each term is of an odd degree. a. Graph the function using a graphing utility. b. Does this graph display any symmetry? c. Evaluate ( ). d. Is an odd function? Explain how you know. Lesson 15: Structure in Graphs of Polynomial Functions S.85

9 Lesson We have talked about -intercepts of the graph of a function in both this lesson and the previous one. The -intercepts correspond to the zeros of the function. Consider the following examples of polynomial functions and their graphs to determine an easy way to find the -intercept of the graph of a polynomial function. ( ) =2 4 3 ( ) = ( ) = Lesson 15: Structure in Graphs of Polynomial Functions S.86

10 Analysis Polynomial PS: SE Practice Fa Construct the rule for a polynomial function f that has the following properties, and evaluate how analytically: a. an x-intercept that bounces on the x-axis b. two x-intercepts that pass through the x-axis c. y-intercept (-1) d. lim f ( x) = + x ± e. decreases on an interval containing x = 0. Then, parametricize f using a nontrivial parameter, and establish its orientation using limits. 2. Construct the rule for a polynomial function f that has the following properties, and evaluate how analytically: a. two x-intercepts that bounce on the x-axis b. R f = (,0] c. y-intercept (-4) d. is even Then, parametricize f using a nontrivial parameter, and establish its orientation using limits.