EVALUATING POLYNOMIAL FUNCTIONS

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1 Page 1 of Evaluating and Graphing Polnomial Functions What ou should learn GOAL 1 Evaluate a polnomial function. GOAL 2 Graph a polnomial function, as applied in Eample 5. Wh ou should learn it To find values of real-life functions, such as the amount of prize mone awarded at the U.S. Open Tennis Tournament in E. 86. REAL LIFE A GOAL 1 polnomial function EVALUATING POLYNOMIAL FUNCTIONS is a function of the form ƒ() = a n n + a n º 1 n º a 1 + a 0 where a n 0, the eponents are all whole numbers, and the coefficients are all real numbers. For this polnomial function, a n is the leading coefficient, a 0 is the constant term, and n is the degree. A polnomial function is in standard form if its terms are written in descending order of eponents from left to right. You are alread familiar with some tpes of polnomial functions. For instance, the linear function ƒ() = is a polnomial function of degree 1. The quadratic function ƒ() = is a polnomial function of degree 2. Here is a summar of common tpes of polnomial functions. Degree Tpe Standard form 0 Constant ƒ() = a 0 1 Linear ƒ() = a 1 + a 0 2 Quadratic ƒ() = a a 1 + a 0 3 Cubic ƒ() = a a a 1 + a 0 4 Quartic ƒ() = a a a a 1 + a 0 E X A M P L E 1 Identifing Polnomial Functions Decide whether the function is a polnomial function. If it is, write the function in standard form and state its degree, tpe, and leading coefficient. a. ƒ() = º 3 4 º 7 b. ƒ() = c. ƒ() = º1 + d. ƒ() = º0.5 + π 2 º 2 a. The function is a polnomial function. Its standard form is ƒ() = º º 7. It has degree 4, so it is a quartic function. The leading coefficient is º3. b. The function is not a polnomial function because the term 3 does not have a variable base and an eponent that is a whole number. c. The function is not a polnomial function because the term 2 º1 has an eponent that is not a whole number. d. The function is a polnomial function. Its standard form is ƒ() = π 2 º 0.5 º 2. It has degree 2, so it is a quadratic function. The leading coefficient is π. 6.2 Evaluating and Graphing Polnomial Functions 329

2 Page 2 of 8 One wa to evaluate a polnomial function is to use direct substitution. For instance, ƒ() = 2 4 º º 7 can be evaluated when = 3 as follows. ƒ(3) = 2(3) 4 º 8(3) 2 + 5(3) º 7 = 162 º º 7 = 98 Another wa to evaluate a polnomial function is to use snthetic substitution. E X A M P L E 2 Using Snthetic Substitution Use snthetic substitution to evaluate ƒ() = 2 4 º º 7 when = 3. Stud Tip In Eample 2, note that the row of coefficients for ƒ() must include a coefficient of 0 for the missing 3 -term. Write the value of and the coefficients of ƒ() as shown. Bring down the leading coefficient. Multipl b 3 and write the result in the net column. Add the numbers in that column and write the sum below the line. Continue to multipl and add, as shown. -value ƒ(3) = (º8 2 ) (º7) º8 5 º7 Coefficients Polnomial in standard form The value of ƒ(3) is the last number ou write, in the bottom right-hand corner. Using snthetic substitution is equivalent to evaluating the polnomial in nested form. ƒ() = º º 7 Write original function. FOCUS ON CAREERS = ( º 8 + 5) º 7 Factor out of first 4 terms. = (( º 8) + 5) º 7 Factor out of first 3 terms. = (((2 + 0) º 8) + 5) º 7 Factor out of first 2 terms. E X A M P L E 3 Evaluating a Polnomial Function in Real Life PHOTOGRAPHER Some photographers work in advertising, some work for newspapers, and some are self-emploed. Others specialize in aerial, police, medical, or scientific photograph. REAL CAREER LINK INTERNET LIFE PHOTOGRAPHY The time t (in seconds) it takes a camera batter to recharge after flashing n times can be modeled b t = n 3 º n n Find the recharge time after 100 flashes. Source: Popular Photograph º º º The recharge time is about 11 seconds. 330 Chapter 6 Polnomials and Polnomial Functions

3 Page 3 of 8 GOAL 2 GRAPHING POLYNOMIAL FUNCTIONS Look Back For help with graphing functions, see pp. 69 and 250. The end behavior of a polnomial function s graph is the behavior of the graph as approaches positive infinit (+ ) or negative infinit (º ). The epression + is read as approaches positive infinit. f() as. f() as. f() as. f() as. f() as. f() as. f() as. f() as ACTIVITY Developing Concepts Investigating End Behavior Use a graphing calculator to graph each function. Then complete these statements: ƒ()? as º and ƒ()? as +. a. ƒ() = 3 b. ƒ() = 4 c. ƒ() = 5 d. ƒ() = 6 e. ƒ() = º 3 f. ƒ() = º 4 g. ƒ() = º 5 h. ƒ() = º 6 How does the sign of the leading coefficient affect the behavior of a polnomial function s graph as +? How is the behavior of a polnomial function s graph as + related to its behavior as º when the function s degree is odd? when it is even? In the activit ou ma have discovered that the end behavior of a polnomial function s graph is determined b the function s degree and leading coefficient. CONCEPT SUMMARY END BEHAVIOR FOR POLYNOMIAL FUNCTIONS The graph of ƒ() = a n n + a n º 1 n º a 1 + a 0 has this end behavior: For a n > 0 and n even, ƒ() + as º and ƒ() + as +. For a n > 0 and n odd, ƒ() º as º and ƒ() + as +. For a n < 0 and n even, ƒ() º as º and ƒ() º as +. For a n < 0 and n odd, ƒ() + as º and ƒ() º as Evaluating and Graphing Polnomial Functions 331

4 Page 4 of 8 E X A M P L E 4 Graphing Polnomial Functions HOMEWORK HELP Visit our Web site for etra eamples. INTERNET Graph (a) ƒ() = º 4 º 1 and (b) ƒ() = º 4 º a. To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. 3 º3 º2 º ƒ() º7 3 3 º1 º The degree is odd and the leading coefficient is positive, so ƒ() º as º and ƒ() + as +. b. To graph the function, make a table of values and plot the corresponding points. Connect the points with a smooth curve and check the end behavior. º3 º2 º ƒ() º21 0 º1 0 3 º16 º The degree is even and the leading coefficient is negative, so ƒ() º as º and ƒ() º as +. REAL LIFE E X A M P L E 5 Graphing a Polnomial Model Biolog A rainbow trout can grow up to 40 inches in length. The weight (in pounds) of a rainbow trout is related to its length (in inches) according to the model = Graph the model. Use our graph to estimate the length of a 10 pound rainbow trout. Make a table of values. The model makes sense onl for positive values of Plot the points and connect them with a smooth curve, as shown at the right. Notice that the leading coefficient of the model is positive and the degree is odd, so the graph rises to the right. Read the graph backwards to see that 27 when = 10. A 10 pound trout is approimatel 27 inches long. Weight (lb) Size of Rainbow Trout Length (in.) 332 Chapter 6 Polnomials and Polnomial Functions

5 Page 5 of 8 GUIDED PRACTICE Vocabular Check Concept Check Skill Check 1. Identif the degree, tpe, leading coefficient, and constant term of the polnomial function ƒ() = 5 º Complete the snthetic substitution shown at the right. Describe each step of the process. 3. Describe the graph of a constant function. Decide whether each function is a polnomial function. If it is, use snthetic substitution to evaluate the function when = º1. 4. ƒ() = 4 5 º 5. ƒ() = º º ƒ() = 6 2 º ƒ() = 14 º Describe the end behavior of the graph of the polnomial function b completing the statements ƒ()? as º and ƒ()? as ƒ() = 3 º 5 9. ƒ() = º 5 º ƒ() = 4 º ƒ() = ƒ() = º ƒ() = º º VIDEO RENTALS The total revenue (actual and projected) from home video rentals in the United States from 1985 to 2005 can be modeled b R = 1.8t 3 º 76t t º2 3 1 º9 2??? 3?? 0 where R is the revenue (in millions of dollars) and t is the number of ears since Graph the function. Source: The Wall Street Journal Almanac PRACTICE AND APPLICATIONS Etra Practice to help ou master skills is on p CLASSIFYING POLYNOMIALS Decide whether the function is a polnomial function. If it is, write the function in standard form and state the degree, tpe, and leading coefficient. 15. ƒ() = 12 º ƒ() = ƒ() = + π 18. ƒ() = º ƒ() = º 3 º2 º ƒ() = º2 21. ƒ() = 2 º ƒ() = 22 º ƒ() = 36 2 º ƒ() = 3 2 º 2 º 25. ƒ() = ƒ() = º6 2 + º 3 HOMEWORK HELP Eample 1: Es Eample 2: Es Eample 3: Es. 81, 82 Eample 4: Es Eample 5: Es DIRECT SUBSTITUTION Use direct substitution to evaluate the polnomial function for the given value of. 27. ƒ() = , = º2 28. ƒ() = 2 3 º º, = ƒ() = , = ƒ() = 2 º 5 + 1, = º1 31. ƒ() = 5 4 º , = ƒ() = º 2 + 5, = º3 33. ƒ() = 11 3 º , = ƒ() = 4 º 2 + 7, = ƒ() = , = ƒ() = º 5 º º, = º2 6.2 Evaluating and Graphing Polnomial Functions 333

6 Page 6 of 8 SYNTHETIC SUBSTITUTION Use snthetic substitution to evaluate the polnomial function for the given value of. 37. ƒ() = , = ƒ() = º º 4 + 8, = ƒ() = º 11, = º5 40. ƒ() = 3 º , = º1 41. ƒ() = º º 5, = ƒ() = º º + 1, = º3 43. ƒ() = º , = º1 44. ƒ() = 3 5 º 2 2 +, = ƒ() = 2 3 º 2 + 6, = ƒ() = º º 4, = º2 END BEHAVIOR PATTERNS Graph each polnomial function in the table. Then cop and complete the table to describe the end behavior of the graph of each function. 47. As As 48. Function º + ƒ() = º5 3?? ƒ() = º 3 + 1?? ƒ() = 2 º 3 3?? ƒ() = 2 2 º 3?? As As Function º + ƒ() = ?? ƒ() = 4 + 2?? ƒ() = 4 º 2 º 1?? ƒ() = 3 4 º 5 2?? MATCHING Use what ou know about end behavior to match the polnomial function with its graph. 49. ƒ() = 4 6 º º ƒ() = º ƒ() = º ƒ() = A. B. C. D. DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the polnomial function b completing these statements: ƒ()? as º and ƒ()? as ƒ() = º ƒ() = º ƒ() = ƒ() = º ƒ() = º º 58. ƒ() = ƒ() = º3 5 º ƒ() = 7 º ƒ() = 3 6 º º ƒ() = 3 8 º ƒ() = º ƒ() = 4 º º Chapter 6 Polnomials and Polnomial Functions

7 Page 7 of 8 GRAPHING POLYNOMIALS Graph the polnomial function. 65. ƒ() = º ƒ() = º ƒ() = ƒ() = 4 º ƒ() = º ƒ() = 2 º ƒ() = 5 º ƒ() = º ƒ() = º ƒ() = º º ƒ() = º ƒ() = 3 º 3 º ƒ() = º 78. ƒ() = 4 º 2 º ƒ() = º º CRITICAL THINKING Give an eample of a polnomial function ƒ such that ƒ() º as º and ƒ() + as SHOPPING The retail space in shopping centers in the United States from 1972 to 1996 can be modeled b S = º0.0068t 3 º 0.27t t where S is the amount of retail space (in millions of square feet) and t is the number of ears since How much retail space was there in 1990? 82. CABLE TELEVISION The average monthl cable TV rate from 1980 to 1997 can be modeled b R = º0.0036t t 2 º 0.073t where R is the monthl rate (in dollars) and t is the number of ears since What was the monthl rate in 1983? FOCUS ON CAREERS NURSING In Eercises 83 and 84, use the following information. From 1985 to 1995, the number of graduates from nursing schools in the United States can be modeled b = º0.036t t 3 º 1.87t 2 º 4.67t where is the number of graduates (in thousands) and t is the number of ears since Source: Statistical Abstract of the United States 83. Describe the end behavior of the graph of the function. From the end behavior, would ou epect the number of nursing graduates in the ear 2010 to be more than or less than the number of nursing graduates in 1995? Eplain. 84. Graph the function for 0 t 10. Use the graph to find the first ear in which there were over 82,500 nursing graduates. NURSE Although the majorit of nurses work in hospitals, nurses also work in doctors offices, in private homes, at nursing homes, and in other communit settings. REAL CAREER LINK INTERNET LIFE TENNIS In Eercises 85 and 86, use the following information. The amount of prize mone for the women s U.S. Open Tennis Tournament from 1970 to 1997 can be modeled b P = 1.141t 2 º 5.837t where P is the prize mone (in thousands of dollars) and t is the number of ears since Source: U.S. Open 85. Describe the end behavior of the graph of the function. From the end behavior, would ou epect the amount of prize mone in the ear 2005 to be more than or less than the amount in 1995? Eplain. 86. Graph the function for 0 t 40. Use the graph to estimate the amount of prize mone in the ear Evaluating and Graphing Polnomial Functions 335

8 Page 8 of 8 Test Preparation Challenge EXTRA CHALLENGE MULTI-STEP PROBLEM To determine whether a Holstein heifer s height is normal, a veterinarian can use the cubic functions L = t 3 º 0.061t t + 30 H = 0.001t 3 º 0.08t t + 31 where L is the minimum normal height (in inches), H is the maimum normal height (in inches), and t is the age (in months). Source: Journal of Dair Science a. What is the normal height range for an 18-month-old Holstein heifer? b. Describe the end behavior of each function s graph. c. Graph the two height functions. d. Writing Suppose a veterinarian eamines a Holstein heifer that is 43 inches tall. About how old do ou think the cow is? How did ou get our answer? EXAMINING END BEHAVIOR Use a spreadsheet or a graphing calculator to evaluate the polnomial functions ƒ() = 3 and g() = 3 º for the given values of. 88. Cop and complete the table. 89. Use the results of Eercise 88 to complete this statement: ƒ( ) As +, g ( )?. Eplain how this statement shows that the functions ƒ and g have the same end behavior as +. A heifer is a oung cow that has not et had calves. ƒ() g() ƒ() g() 50??? 100??? 500??? 1000??? 5000??? MIXED REVIEW SIMPLIFYING EXPRESSIONS Simplif the epression. (Review 1.2 for 6.3) º 2 º º º º ( 2 + 2) ( º 4) º º 2 º º 1 º (2 2 + º 3) STANDARD FORM Write the quadratic function in standard form. (Review 5.1 for 6.3) 96. = º4( º 2) = º2( + 6)( º 5) 98. = 2( º 7)( + 4) 99. = 4( º 3) 2 º = º( + 5) = º3( º 5) SOLVING QUADRATIC EQUATIONS Solve the equation. (Review 5.4) = º = º º = = = = º 4 = º3 2 º = = º Chapter 6 Polnomials and Polnomial Functions