6.2 Evaluating and Graphing Polynomial Functions

1 Page 1 of 2 Evaluating and GraphingPolynomial FunctionsEVALUATINGPOLYNOMIALFUNCTIONSA is a function of the form (x) = anxn+ an 1xn 1+ ..+ a1x+ a0where an 0, the exponents are all whole numbers, and the coefficients are all realnumbers. For this Polynomial function, anis the a0is theand nis the A Polynomial function is in if its terms are written in descending order of exponents from left to are already familiar with some types of Polynomial Functions . For instance, thelinear function (x) = 3x+ 2 is a Polynomial function of degree 1. The quadraticfunction (x) = x2+ 3x+ 2 is a Polynomial function of degree 2. Here is a summaryof common types of Polynomial Polynomial FunctionsDecide whether the function is a Polynomial function. If it is, write the function instandard form and state its degree, type, and leading (x) = 12 x2 3x4 7b.

2 (x) = x3+ 3xc. (x) = 6x2+ 2x 1+ xd. (x) = + x2 2 function is a Polynomial function. Its standard form is (x)= 3x4+ 21 x2 has degree 4, so it is a quartic function. The leading coefficient is function is not a Polynomial function because the term 3xdoes not have avariable base and an exponent that is a whole function is not a Polynomial function because the term 2x 1has an exponent thatis not a whole function is a Polynomial function. Its standard form is (x)= x2 2 .It has degree 2, so it is a quadratic function. The leading coefficient is .EXAMPLE 1standard term,leading coefficient, Polynomial and Graphing Polynomial Functions329 Evaluate apolynomial a polynomialfunction, as applied inExample 5. To find values of real-lifefunctions, such as theamount of prize moneyawarded at the OpenTennis Tournament in Ex.

3 You should learn itGOAL2 GOAL1 What you should form0 Constant (x) = a01 Linear (x) = a1x+ a02 Quadratic (x) = a2x2+ a1x+ a03 Cubic (x) = a3x3+ a2x2+ a1x+ a04 Quartic (x) = a4x4+ a3x3+ a2x2+ a1x+ a0 Page 1 of 2330 Chapter 6 polynomials and Polynomial FunctionsOne way to evaluate a Polynomial function is to use direct substitution. For instance, (x) = 2x4 8x2+ 5x 7 can be evaluated when x= 3as follows. (3) = 2(3)4 8(3)2+ 5(3) 7= 162 72 + 15 7= 98 Another way to evaluate a Polynomial function is to use Using Synthetic SubstitutionUse synthetic substitution to evaluate (x) = 2x4 8x2+ 5x 7 when x= the value of xand the coefficients of (x) as shown. Bring down the leadingcoefficient. Multiply by 3and write the result in the next column.

4 Addthe numbers inthat column and write the sum below the line. Continue to multiply and add, as shown. (3) = 98.. Using synthetic substitution is equivalent to Evaluating the Polynomial in nested form. (x) = 2x4+0x3 8x2+ 5x 7 Write original (2x3+0x2 8x+ 5)x 7 Factor xout of first 4 ((2x2+0x 8)x+ 5)x 7 Factor xout of first 3 (((2x+0)x 8)x+5)x 7 Factor xout of first 2 a Polynomial Function in Real LifePHOTOGRAPHYThe time t(in seconds) it takes a camera battery to recharge afterflashing ntimes can be modeled by t= + + Findthe recharge time after 100 flashes. Source: Popular PhotographySOLUTION The recharge time is about 11 3 EXAMPLE 2synthetic photographerswork in advertising, somework for newspapers, andsome are specialize in aerial,police, medical, or ONCAREERSx-value2x4+ 0x3 +( 8x2)+ 5x+ ( 7) Polynomial in standard form320 8 5 7 Coefficients618 30 1052610 3598 The value of (3) is the lastnumber you write, in thebottom right-hand corner.

5 Study TipIn Example 2, note thatthe row of coefficients for (x) must include acoefficient of 0 for the missing 1 of 2 GRAPHINGPOLYNOMIALFUNCTIONSThe of a Polynomial function s graph is the behavior of the graph as xapproaches positive infinity (+ ) or negative infinity ( ). The expression x + is read as xapproaches positive infinity. f(x) as x .f(x) as x .yxxf(x) as x .f(x) as x .yf(x) as x .f(x) as x .xyf(x) as x .f(x) as x .yxend behaviorGOAL2 Investigating End BehaviorUse a Graphing calculator to graph each function. Then complete thesestatements: (x) ?as x and (x) ?as x + .a. (x)=x3b. (x)=x4c. (x)=x5d. (x)=x6e. (x)= x3f. (x)= x4g. (x)= x5h. (x)= x6 How does the sign of the leading coefficient affect the behavior of apolynomial function s graph as x + ?

6 How is the behavior of a Polynomial function s graph as x + related toits behavior as x when the function s degree is odd? when it is even?321 DevelopingConceptsACTIVITYThe graph of (x) = anxn+ an 1xn 1+.. + a1x+ a0has this end behavior: For an> 0 and neven, (x) + as x and (x) + asx + . For an> 0 and nodd, (x) as x and (x) + as x + . For an< 0 and neven, (x) as x and (x) as x + . For an< 0 and nodd, (x) + as x and (x) as x + .END BEHAVIOR FOR Polynomial FUNCTIONSCONCEPTSUMMARYIn the activity you may have discovered that the end behavior of a polynomialfunction s graph is determined by the function s degree and leading and Graphing Polynomial Functions331 STUDENTHELPLook Back For help with graphingfunctions, see pp.

7 69and 1 of 2 Graphing Polynomial FunctionsGraph (a) (x) = x3+x2 4x 1 and (b) (x) = x4 2x3+ 2x2+ graph the function, make a table of values and plot thecorresponding points. Connect the points with a smooth curve and check the end degree is odd and the leading coefficient is positive, so (x) as x and (x) + as x + . graph the function, make a table of values and plot thecorresponding points. Connect the points with a smooth curve and check the end degree is even and the leading coefficient is negative, so (x) as x and (x) as x + . Graphing a Polynomial ModelA rainbow trout can grow up to 40 inches in length. The weight y(in pounds) of a rainbow trout is related to its length x(in inches) according to the modely= Graph the model.

8 Use your graph to estimate the length of a 10 pound rainbow table of values. The model makes sense only for positive values of points and connect them with asmooth curve, as shown at the right. Noticethat the leading coefficient of the model ispositive and the degree is odd, so the graphrises to the right. Readthe graph backwards to see that x 27when y= 10. A 10 pound trout is approximately27 inches 5 EXAMPLE 4332 Chapter 6 polynomials and Polynomial Functionsyx13yx11 Size of Rainbow TroutLength (in.)Weight (lb)0y102030400301020xREALLIFEREALLIFEB iologyx 3 2 10123 (x) 7 33 1 3323x 3 2 10123 (x) 21 0 103 16 105x0510152025303540y0 32 HOMEWORK HELPV isit our Web extra 1 of and Graphing Polynomial the degree, type, leading coefficient, and constant term of thepolynomial function (x) = 5x the synthetic substitution shown atthe right.

9 Describe each step of the the graph of a constant whether each function is a Polynomial function. If it is, use syntheticsubstitution to evaluate the function when x= (x) = x4 5 x5. (x) = x3+ x2 x 3+ 36. (x) = 62x 12x7. (x) = 14 21x2+5x4 Describe the end behavior of the graph of the Polynomial function bycompleting the statements (x) ?as x and (x) ?as x + .8. (x)= x3 5x9. (x) = x5 3x3+ 210. (x) = x4 4x2+ x11. (x) = x+1212. (x) = x2+3x+113. (x) = x8+9x5 total revenue (actual and projected) from home videorentals in the United States from 1985 to 2005 can be modeled by R= 76t2+ 1099t+ 2600where Ris the revenue (in millions of dollars) and tis the number of years since1985. Graph the function. Source: The Wall Street Journal AlmanacCLASSIFYINGPOLYNOMIALSD ecide whether the function is a polynomialfunction.

10 If it is, write the function in standard form and state the degree, type,and leading (x) = 12 5x16. (x) = 2x+ 35 x4+ 917. (x) = x+ 18. (x) = x2 2 +x 519. (x) = x 3x 2 2x320. (x) = 221. (x) = x2 x+ 122. (x) = 22 19x+2x23. (x) = 36x2 x3+ x424. (x)=3x2 2x x25. (x)=3x326. (x)= 6x2+x DIRECTSUBSTITUTIONUse direct substitution to evaluate the polynomialfunction for the given value of (x) = 2x3+ 5x2+ 4x+ 8, x= 228. (x) = 2x3 x4+ 5x2 x, x= 329. (x) = x+ 12 x3, x= 430. (x) = x2 x5+ 1, x= 131. (x) = 5x4 8x3+7x2, x=132. (x) = x3+3x2 2x+5, x= 333. (x) = 11x3 6x2+2, x=034. (x) = x4 2x+ 7, x= 235. (x)=7x3+9x2+3x, x=1036. (x)= x5 4x3+6x2 x, x= 23 xPRACTICEANDAPPLICATIONSGUIDEDPRACTICE 231 92???3? ?0 Vocabulary Check Concept Check Skill Check STUDENTHELPE xtra Practice to help you masterskills is on p.