Evaluate and Graph Polynomial Functions

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1 5.2 Evaluate and Graph Polnomial Functions Before You evaluated and graphed linear and quadratic functions. Now You will evaluate and graph other polnomial functions. Wh? So ou can model skateboarding participation, as in E. 55. Ke Vocabular polnomial polnomial function snthetic substitution end behavior Recall that a monomial is a number, a variable, or a product of numbers and variables. A polnomial is a monomial or a sum of monomials. A polnomial function is a function of the form f() 5 a n n a n 2 n 2... a a 0 where a n Þ 0, the eponents are all whole numbers, and the coefficients are all real numbers. For this function, a n is the leading coefficient, n is the degree, and a 0 is the constant term. A polnomial function is in standard form if its terms are written in descending order of eponents from left to right. Common Polnomial Functions Degree Tpe Standard form Eample 0 Constant f() 5 a 0 f() 5 24 Linear f() 5 a a 0 f() Quadratic f() 5 a 2 2 a a 0 f() Cubic f() 5 a 3 3 a 2 2 a a 0 f() Quartic f() 5 a 4 4 a 3 3 a 2 2 a a 0 f() E XAMPLE Identif polnomial functions Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. a. h() } b. g() Ï } 3 π 2 c. f() d. k() Solution a. The function is a polnomial function that is alread written in standard form. It has degree 4 (quartic) and a leading coefficient of. b. The function is a polnomial function written as g() 5 π Ï } 3 in standard form. It has degree 2 (quadratic) and a leading coefficient of π. c. The function is not a polnomial function because the term 3 2 has an eponent that is not a whole number. d. The function is not a polnomial function because the term 2 does not have a variable base and an eponent that is a whole number. 5.2 Evaluate and Graph Polnomial Functions 337

2 E XAMPLE 2 Evaluate b direct substitution Use direct substitution to evaluate f() when 5 3. f() Write original function. f(3) 5 2(3) 4 2 5(3) 3 2 4(3) 8 Substitute 3 for Evaluate powers and multipl Simplif. GUIDED PRACTICE for Eamples and 2 Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient.. f() p() h() π 2 3 Use direct substitution to evaluate the polnomial function for the given value of. 4. f() ; g() ; 5 4 SYNTHETIC SUBSTITUTION Another wa to evaluate a polnomial function is to use snthetic substitution. This method, shown in the net eample, involves fewer operations than direct substitution. E XAMPLE 3 Evaluate b snthetic substitution Use snthetic substitution to evaluate f() from Eample 2 when 5 3. AVOID ERRORS The row of coefficients for f() must include a coefficient of 0 for the missing 2 -term. Solution STEP Write the coefficients of f() in order of descending eponents. Write the value at which f() is being evaluated to the left. -value coefficients STEP 2 Bring down the leading coefficient. Multipl the leading coefficient b the -value. Write the product under the second coefficient. Add STEP 3 2 Multipl the previous sum b the -value. Write the product under the third coefficient. Add. Repeat for all of the remaining coefficients. The final sum is the value of f() at the given -value c Snthetic substitution gives f(3) 5 23, which matches the result in Eample Chapter 5 Polnomials and Polnomial Functions

3 END BEHAVIOR The end behavior of a function s graph is the behavior of the graph as approaches positive infinit (`) or negative infinit (2`). For the graph of a polnomial function, the end behavior is determined b the function s degree and the sign of its leading coefficient. KEY CONCEPT For Your Notebook End Behavior of Polnomial Functions READING The epression ` is read as approaches positive infinit. Degree: odd Leading coefficient: positive f() 2` as 2` f() ` as ` Degree: odd Leading coefficient: negative f() ` as 2` f() 2` as ` Degree: even Leading coefficient: positive Degree: even Leading coefficient: negative f() ` as 2` f() ` as ` f() 2` as 2` f() 2` as ` E XAMPLE 4 Standardized Test Practice What is true about the degree and leading coefficient of the polnomial function whose graph is shown? A Degree is odd; leading coefficient is positive 4 B Degree is odd; leading coefficient is negative C Degree is even; leading coefficient is positive 3 D Degree is even; leading coefficient is negative From the graph, f() 2` as 2` and f() 2` as `. So, the degree is even and the leading coefficient is negative. c The correct answer is D. A B C D GUIDED PRACTICE for Eamples 3 and 4 Use snthetic substitution to evaluate the polnomial function for the given value of. 6. f() ; g() ; Describe the degree and leading coefficient of the polnomial function whose graph is shown. 5.2 Evaluate and Graph Polnomial Functions 339

4 GRAPHING POLYNOMIAL FUNCTIONS To graph a polnomial function, first plot points to determine the shape of the graph s middle portion. Then use what ou know about end behavior to sketch the ends of the graph. E XAMPLE 5 Graph polnomial functions Graph (a) f() and (b) f() Solution 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. (22, 3) (, 0) (2, 2) The degree is odd and leading coefficient is negative. So, f() ` as 2` and f() 2` as `. (2, 24) (0, 23) 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 (0, 4) (2, 2) (, 0) The degree is even and leading coefficient is positive. So, f() ` as 2` and f() ` as `. (2, 24) at classzone.com E XAMPLE 6 Solve a multi-step problem PHYSICAL SCIENCE The energ E (in foot-pounds) in each square foot of a wave is given b the model E s 4 where s is the wind speed (in knots). Graph the model. Use the graph to estimate the wind speed needed to generate a wave with 000 foot-pounds of energ per square foot. Solution STEP STEP 2 Make a table of values. The model onl deals with positive values of s. s E Plot the points and connect them with a smooth curve. Because the leading coefficient is positive and the degree is even, the graph rises to the right. Energ per square foot (foot-pounds) STEP 3 Eamine the graph to see that s < 24 when E c The wind speed needed to generate the wave is about 24 knots. E Wave Energ (24, 000) s Wind speed (knots) 340 Chapter 5 Polnomials and Polnomial Functions

5 GUIDED PRACTICE for Eamples 5 and 6 Graph the polnomial function. 9. f() f() f() WHAT IF? If wind speed is measured in miles per hour, the model in Eample 6 becomes E s 4. Graph this model. What wind speed is needed to generate a wave with 2000 foot-pounds of energ per square foot? 5.2 EXERCISES SKILL PRACTICE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 2, 27, and 57 5 STANDARDIZED TEST PRACTICE Es. 2, 24, 37, 50, 52, and 59 5 MULTIPLE REPRESENTATIONS E. 56. VOCABULARY Identif the degree, tpe, leading coefficient, and constant term of the polnomial function f() WRITING Eplain what is meant b the end behavior of a polnomial function. EXAMPLE on p. 337 for Es. 3 8 POLYNOMIAL FUNCTIONS Decide whether the function is a polnomial function. If so, write it in standard form and state its degree, tpe, and leading coefficient. 3. f() f() g() 5 π 4 Ï } 6 6. h() 5 3 Ï } h() 5 2 } g() } 2 EXAMPLE 2 on p. 338 for Es. 9 4 DIRECT SUBSTITUTION Use direct substitution to evaluate the polnomial function for the given value of. 9. f() ; f() ; 5 2. g() ; h() ; h() 5 } } 4 3 0; g() ; 5 22 EXAMPLE 3 on p. 338 for Es SYNTHETIC SUBSTITUTION Use snthetic substitution to evaluate the polnomial function for the given value of. 5. f() ; f() ; g() ; h() ; f() ; g() ; h() ; f() ; ERROR ANALYSIS Describe and correct the error in evaluating the polnomial function f() when Evaluate and Graph Polnomial Functions 34

6 EXAMPLE 4 on p. 339 for Es MULTIPLE CHOICE The graph of a polnomial function is shown. What is true about the function s degree and leading coefficient? A The degree is odd and the leading coefficient is positive. B The degree is odd and the leading coefficient is negative. 2 C The degree is even and the leading coefficient is positive. D The degree is even and the leading coefficient is negative. USING END BEHAVIOR Describe the degree and leading coefficient of the polnomial function whose graph is shown DESCRIBING END BEHAVIOR Describe the end behavior of the graph of the polnomial function b completing these statements: f()? as 2` and f()? as `. 28. f() f() f() f() f() f() f() f() f() OPEN-ENDED MATH Write a polnomial function f of degree 5 such that the end behavior of the graph of f is given b f() ` as 2` and f() 2` as `. Then graph the function to verif our answer. EXAMPLE 5 on p. 340 for Es GRAPHING POLYNOMIALS Graph the polnomial function. 38. f() f() f() f() f() f() f() f() f() f() f() f() MULTIPLE CHOICE Which function is represented b the graph shown? 2 A f() 5 } 3 3 B f() 5 2 } 3 3 C f() 5 } D f() 5 2 } VISUAL THINKING Suppose f() ` as 2` and f() 2` as `. Describe the end behavior of g() 5 2f(). 52. SHORT RESPONSE A cubic polnomial function f has leading coefficient 2 and constant term 25. If f() 5 0 and f(2) 5 3, what is f(25)? Eplain how ou found our answer. 5 WORKED-OUT SOLUTIONS 342 Chapter 5 Polnomials on p. WS and Polnomial Functions 5 STANDARDIZED TEST PRACTICE 5 MULTIPLE REPRESENTATIONS

7 53. CHALLENGE Let f() 5 3 and g() a. Cop and complete the table. b. Use the numbers in the table to complete this statement: As `, } f() g()?. c. Eplain how the result from part (b) shows that the functions f and g have the same end behavior as `. f () g() f() } g() 0??? 20??? 50??? 00??? 200??? PROBLEM SOLVING EXAMPLE 6 on p. 340 for Es DIAMONDS The weight of an ideal round-cut diamond can be modeled b w d d d where w is the diamond s weight (in carats) and d is its diameter (in millimeters). According to the model, what is the weight of a diamond with a diameter of 5 millimeters? 55. SKATEBOARDING From 992 to 2003, the number of people in the United States who participated in skateboarding can be modeled b S t 4 0.4t t t 5.5 where S is the number of participants (in millions) and t is the number of ears since 992. Graph the model. Then use the graph to estimate the first ear that the number of skateboarding participants was greater than 8 million. 56. MULTIPLE REPRESENTATIONS From 987 to 2003, the number of indoor movie screens M in the United States can be modeled b M 5 2.0t 3 267t t 2,600 where t is the number of ears since 987. a. Classifing a Function State the degree and tpe of the function. b. Making a Table Make a table of values for the function. c. Sketching a Graph Use our table to graph the function. 57. SNOWBOARDING From 992 to 2003, the number of people in the United States who participated in snowboarding can be modeled b S t t t t.2 where S is the number of participants (in millions) and t is the number of ears since 992. Graph the model. Use the graph to estimate the first ear that the number of snowboarding participants was greater than 2 million. 5.2 Evaluate and Graph Polnomial Functions 343

8 58. MULTI-STEP PROBLEM From 980 to 2002, the number of quarterl periodicals P published in the United States can be modeled b P t t t t 450 where t is the number of ears since 980. a. Describe the end behavior of the graph of the model. b. Graph the model on the domain 0 t 22. c. Use the model to predict the number of quarterl periodicals in the ear 200. Is it appropriate to use the model to make this prediction? Eplain. 59. EXTENDED RESPONSE The weight of Sarus crane chicks S and hooded crane chicks H (both in grams) during the 0 das following hatching can be modeled b the functions S t t t 36 H t 3 3.7t t 24 where t is the number of das after hatching. a. Calculate According to the models, what is the difference in weight between 5-da-old Sarus crane chicks and hooded crane chicks? b. Graph Sketch the graphs of the two models. c. Appl A biologist finds that the weight of a crane chick after 3 das is 30 grams. What species of crane is the chick more likel to be? Eplain how ou found our answer. 60. CHALLENGE The weight (in pounds) of a rainbow trout can be modeled b where is the length of the trout (in inches). a. Write a function that relates the weight and length of a rainbow trout if is measured in kilograms and is measured in centimeters. Use the fact that kilogram ø 2.20 pounds and centimeter ø inch. b. Graph the original function and the function from part (a) in the same coordinate plane. What tpe of transformation can ou appl to the graph of to produce the graph from part (a)? MIXED REVIEW Solve the equation or inequalit. 6. 2b b (p. 8) n (p. 8) < 6 2 < 5 (p. 4) (p. 252) q q 5 2 (p. 259) 66. z 2 5z < 36 (p. 300) The variables and var directl. Write an equation that relates and. Then find the value of when (p. 07) , , , , , , PREVIEW Prepare for Lesson 5.3 in Es Write the quadratic function in standard form. (p. 245) ( 3)( 2 7) ( 2 4)( 2) ( 2 5) ( 2 6) } 2 ( 2 4) } 5 ( 4)( 9) Chapter 5 EXTRA Polnomials PRACTICE and Polnomial for Lesson Functions 5.2, p. 04 ONLINE QUIZ at classzone.com

9 Use after Lesson Set a Good Viewing Window classzone.com Kestrokes QUESTION What is a good viewing window for a polnomial function? When ou graph a function with a graphing calculator, ou should choose a viewing window that displas the important characteristics of the graph. EXAMPLE Graph a polnomial function Graph f() STEP Graph the function Graph the function in the standard viewing window. STEP 2 Adjust horizontall Adjust the horizontal scale so that the end behavior of the graph as ` is visible. STEP 3 Adjust verticall Adjust the vertical scale so that the turning points and end behavior of the graph as 2` are visible. 20 0, , , P RACTICE Find intervals for and that describe a good viewing window for the graph of the polnomial function.. f() f() f() f() f() f() f() f() REASONING Let g() 5 f() c where f() and g() are polnomial functions and c is a positive constant. How is a good viewing window for the graph of f() related to a good viewing window for the graph of g()? 0. BASEBALL From 994 to 2003, the average salar S (in thousands of dollars) for major league baseball plaers can be modeled b S() where is the number of ears since 994. Find intervals for the horizontal and vertical aes that describe a good viewing window for the graph of S. 5.2 Evaluate and Graph Polnomial Functions 345