1. P OLYNOMIAL F UNCTIONS OF H IGHER D EGREE Section 2.2

2. O BJECTIVES : Use the Leading Coefficient Test to determine the end behavior of graphs of polynomial functions. Find and use zeros of polynomial functions. Use the Intermediate Value Theorem to help locate zeros of polynomial functions.

3. H IGHER D EGREE P OLYNOMIAL F UNCTIONS AND G RAPHS a n is called the leading coefficient a n x n is called the dominating term a 0 is called the constant term P (0) = a 0 is the y -intercept of the graph of P Polynomial Function A polynomial function of degree n in the variable x is a function defined by where each a i is real, a n  0, and n is a whole number.

4. G RAPHS OF P OLYNOMIAL F UNCTIONS Graphs of polynomial functions are continuous: there are no breaks, holes, or gaps the graphs are smooth and rounded: there are no sharp turns Sample polynomial graphs: f(x) = x n

5. E XAMPLES OF P OLYNOMIAL F UNCTIONS

6. E XAMPLES OF N ONPOLYNOMIAL F UNCTIONS

7. G RAPHS OF P OLYNOMIAL F UNCTIONS The domain of every polynomial function is: Polynomial functions are continuous over their domain (entire graph can be drawn without lifting pencil) Even polynomial functions,, are symmetric with respect to y-axis Odd polynomial functions,, are symmetric with respect to origin Note: Many polynomial functions are neither even nor odd

8. C UBIC F UNCTIONS : O DD D EGREE P OLYNOMIALS The cubic function is a third degree polynomial of the form In general, the graph of a cubic function will resemble one of the following shapes.

9. Q UARTIC F UNCTIONS : E VEN D EGREE P OLYNOMIALS The quartic function is a fourth degree polynomial of the form In general, the graph of a quartic function will resemble one of the following shapes. The dashed portions indicate irregular, but smooth, behavior.

10. E VEN - AND O DD -D EGREE F UNCTIONS

11. T HE L EADING C OEFFICIENT T EST Let ax n be the dominating term of a polynomial function P. n odd 1.If a > 0, the graph of P falls on the left and rises on the right. 2.If a < 0, the graph of P rises on the left and falls on the right. n even 1.If a > 0, the graph of P opens up. 2.If a < 0, the graph of P opens down.

12. T HE L EADING C OEFFICIENT T EST

13. U SING T HE L EADING C OEFFICIENT T EST Match each function with its graph. Solution : f matches C, g matches A, h matches B, k matches D. A. B. C.D.

14. Y OUR T URN : Using the leading term-test, match each of the following functions with one of the graphs A  D, which follow. 1)2) 3)4)

15. S OLUTION CNegativeEven 4)  x 6 APositiveOdd3) x 5 BNegativeOdd 2)  5 x 3 DPositiveEven1) 3 x 4 GraphSign of Leading Coeff. Degree of Leading Term Leading Term

16. M AXIMUMS AND M INIMUMS Turning points – where the graph of a function changes from increasing to decreasing or vice versa Local maximum point – highest point or “peak” in an interval function values at these points are called local maxima Local minimum point – lowest point or “valley” in an interval function values at these points are called local minima Extrema – plural of extremum, includes all local maxima and local minima

17. T URNING P OINTS OF P OLYNOMIAL F UNCTIONS The point at which the graph of a polynomial function changes from increasing to decreasing or vice versa is called a turning point of the function

18. N UMBER OF L OCAL E XTREMA A linear function has degree 1 and no local extrema. A quadratic function has degree 2 with one extreme point. A cubic function has degree 3 with at most two local extrema. A quartic function has degree 4 with at most three local extrema. Extending this idea: Number of Turning Points The number of turning points of the graph of a polynomial function of degree n  1 is at most n – 1.

19. T URNING P OINTS OF P OLYNOMIAL F UNCTIONS A polynomial function of degree n has at most n – 1 turning points with at least one turning point between successive zeros Given that the following function has zeros at 1 and -1, what do you know about its turning points? It has at most how many? Where is at least one of those located?

20. I DENTIFYING L OCAL AND A BSOLUTE E XTREMA Example Consider the following graph. (a) Name and classify the local extrema of f. (b) Name and classify the local extrema of g. (c) Describe the absolute extrema for f and g. Local Min points: ( a, b ),( e, h ); Local Max point: ( c, d ) Local Min point: ( j, k ); Local Max point: ( m, n ) f has an absolute minimum value of h, but no absolute maximum. g has no absolute extrema.

21. R EVIEW E ND B EHAVIOR – T HE L EADING C OEFFICIENT T EST End behavior of graphs of polynomial functions is determined by characteristics of highest degree term: If degree of polynomial is odd and coefficient of highest degree is positive, ends look like: If degree of polynomial is odd and coefficient of highest degree is negative, ends look like: If degree of polynomial is even and coefficient of highest degree is positive, ends look like: If degree of polynomial is even and coefficient of highest degree is negative, ends look like:

22. X -I NTERCEPTS (R EAL Z EROS ) Example Find the x- intercepts of Solution By using the graphing calculator in a standard viewing window, the x- intercepts (real zeros) are –2, approximately –3.30, and approximately.30. Number Of x -Intercepts of a Polynomial Function A polynomial function of degree n will have a maximum of n x- intercepts (real zeros).

23. R EAL Z EROS OF P OLYNOMIAL F UNCTIONS

24. E XAMPLE Find all zeros of f(x)= x 3 +4x 2 -3x-12. By definition, the zeros are the values of x for which f(x) is equal to 0. So, the real zeros are x=-4 and x= ±√3, and the corresponding x-intercepts are (-4, 0), ( √ 3, 0) and (- √ 3, 0).

25. Y OUR T URN : Find all zeros of x 3 +2x 2 - 4x-8=0. Solution: x = 2, -2, -2 Find all the real zeros of f(x) =x 3 -12x 2 +36x Solution: x = 0, 6, 6

26. M ULTIPLICITY OF A Z ERO The multiplicity of the zero refers to the number of times a zero appears. e.g. – x = 0 leads to a single zero – ( x + 2) 2 leads to a zero of –2 with multiplicity two – ( x – 1) 3 leads to a zero of 1 with multiplicity three

27. M ULTIPLICITIES OF Z EROS Observe the behavior around the zeros of the polynomials The following figure illustrates some conclusions.

28. R EPEATED Z EROS

29. E XAMPLE y = (x + 2)²(x − 1)³ Answer. −2 is a root of multiplicity 2, and 1 is a root of multiplicity 3. These are the 5 roots: −2, −2, 1, 1, 1.

30. Y OUR T URN : y = x³(x + 2) 4 (x − 3) 5 Answer. 0 is a root of multiplicity 3, -2 is a root of multiplicity 4, and 3 is a root of multiplicity 5.

31. T RUE OR F ALSE ? 1.) The function must have 1 real zero. 2.) The function has no real zeros. 3.) An odd degree polynomial function must have at least 1 real zero. 4.) An even degree polynomial function must have at least 1 real zero. True False

32. C OMPREHENSIVE G RAPHS The most important features of the graph of a polynomial function are: 1. intercepts, 2. extrema, 3. end behavior. A comprehensive graph of a polynomial function will exhibit the following features: 1. all x -intercepts (if any), 2. the y -intercept, 3. all extreme points (if any), 4. enough of the graph to exhibit end behavior.

33. E XAMPLE 1 Use the graph of the polynomial function f shown. a) How many turning points and x -intercepts are there? b) Is the leading coefficient a positive or negative? Is the degree odd or even? c) Determine the minimum degree of f.

34. E XAMPLE 1 Solution a) There are four turning points corresponding to the two “hills” and two “valleys”. There appear to be 4 x-intercepts.

35. E XAMPLE 1 Solution continued b) b) The left side rises and the right side falls. Therefore, a < 0 and the polynomial function has odd degree. c)The graph has four turning points. A polynomial of degree n can have at most n  1 turning points. Therefore, f must be at least degree 5.

36. S TEPS TO G RAPH A P OLYNOMIAL F UNCTION 1. Use the leading-term test to determine the end behavior. 2.Find the zeros of the function by solving f ( x ) = 0. Any real zeros are the first coordinates of the x -intercepts. 3.Use the x -intercepts (zeros) to divide the x -axis into intervals and choose a test point in each interval to determine the sign of all function values in that interval. 4.Find f (0). This gives the y -intercept of the function. 5.If necessary, find additional function values to determine the general shape of the graph and then draw the graph. 6.As a partial check, use the facts that the graph has at most n x -intercepts and at most n  1 turning points. Multiplicity of zeros can also be considered in order to check where the graph crosses or is tangent to the x -axis.

37. E XAMPLE Graph the polynomial function f ( x ) = 2 x 3 + x 2  8 x  4. Solution: 1. The leading term is 2 x 3. The degree, 3, is odd, the coefficient, 2, is positive. Thus the end behavior of the graph will appear as: 2.To find the zero, we solve f ( x ) = 0. Here we can use factoring by grouping.

38. E XAMPLE CONTINUED Factor: The zeros are  1/2,  2, and 2. The x -intercepts are (  2, 0), (  1/2, 0), and (2, 0). 3.The zeros divide the x -axis into four intervals: ( ,  2),(  2,  1/2),(  1/2, 2), and (2,  ). We choose a test value for x from each interval and find f ( x ).

39. E XAMPLE CONTINUED Above x -axis+353 (2,  ) Below x -axis 99 1 (  1/2, 2) Above x -axis+3 11(  2,  1/2) Below x -axis  25 33( ,  2) Location of points on graph Sign of f ( x )Function value, f ( x ) Test Value, x Interval 4. To determine the y -intercept, we find f (0): The y -intercept is (0,  4).

40. E XAMPLE CONTINUED 5.We find a few additional points and complete the graph. 6.The degree of f is 3. The graph of f can have at most 3 x - intercepts and at most 2 turning points. It has 3 x - intercepts and 2 turning points. Each zero has a multiplicity of 1; thus the graph crosses the x - axis at  2,  1/2, and 2. The graph has the end behavior described in step (1). The graph appears to be correct. 77 1.5 3.5  1.5 99  2.5 f(x)f(x) x

41. Y OUR T URN : Solution: The dominating term is –2 x 5, so the end behavior will rise on the left and fall on the right. Because –4 and 1 are x- intercepts determined by zeros of even multiplicity, the graph will be tangent to the x -axis at these x- intercepts. The y -intercept is –96. Interv al Test xf(x)Sign f(x) (- ∞, -4) -5144+ (-4, -3)-3.55.06+ (-3, 1)-144- (1, ∞ ) 2-360-

42. Example: Sketch the graph of f (x) = 4x 2 – x 4. 1. Write the polynomial function in standard form: f (x) = –x 4 + 4x 2 The leading coefficient is negative and the degree is even. 2. Find the zeros of the polynomial by factoring. f (x) = –x 4 + 4x 2 = –x 2 (x 2 – 4) = – x 2 (x + 2)(x –2) Zeros: x = –2, 2 multiplicity 1 x = 0 multiplicity 2 x-intercepts: (–2, 0), (2, 0) crosses through (0, 0) touches only Example continued as, x y (2, 0) (0, 0) (–2, 0)

43. Example continued: Sketch the graph of f (x) = 4x 2 – x 4. x y

44. Y OUR T URN : Sketch: f (x) = x 4 – x 3 – 2x 2 Factor completely: f (x) = x 4 – x 3 – 2x 2 = x 2 (x + 1)(x – 2). The real zeros are x = –1, x = 0 (double zero), and x = 2. These correspond to the x-intercepts (–1, 0), (0, 0) and (2, 0). y x

45. I NTERMEDIATE V ALUE T HEOREM

46. W HAT THIS MEANS In other words, if one point is above the x-axis and the other point is below the x-axis, then because P(x) is continuous and will have to cross the x-axis to connect the two points, P(x) must have a zero somewhere between a and b.

47. E XAMPLE Using the intermediate value theorem, determine, if possible, whether the function has a real zero between a and b. a) f ( x ) = x 3 + x 2  8 x ; a =  4 b =  1 b) f ( x ) = x 3 + x 2  8 x ; a = 1 b = 3

48. S OLUTION We find f ( a ) and f ( b ) and determine where they differ in sign. The graph of f ( x ) provides a visual check. f (  4) = (  4) 3 + (  4) 2  8(  4) =  16 f (  1) = (  1) 3 + (  1) 2  8(  1) = 8 By the intermediate value theorem, since f (  4) and f (  1) have opposite signs, then f ( x ) has a zero between  4 and  1. f (1) = (1) 3 + (1) 2  8(1) =  6 f (3) = (3) 3 + (3) 2  8(3) = 12 By the intermediate value theorem, since f (1) and f (3) have opposite signs, then f ( x ) has a zero between 1 and 3. a) f ( x ) = x 3 + x 2  8 x ; a =  4 b =  1 b) f ( x ) = x 3 + x 2  8 x ; a = 1 b = 3

49. Show that the polynomial function f(x)=x 3 - 2x+9 has a real zero between - 3 and - 2. Y OUR T URN :

50. P OLYNOMIAL F UNCTION S ATISFYING G IVEN C ONDITIONS Example Find a polynomial function with real coefficients of lowest possible degree having a zero 2 of multiplicity 2, a zero 0 of multiplicity 3, and a zero of -3. Solution This is one of many such functions. Multiplying P ( x ) by any nonzero number will yield another function satisfying these conditions.

51. Y OUR T URN : Find a polynomial function with real coefficients of lowest possible degree having a zero -1 of multiplicity 2, a zero -2 of multiplicity 2, and a zero of 0.

52. A SSIGNMENT Section 2.2, pg. 112-115: #1-9 all, 15-39 odd, 49- 63 odd, 65-77 odd, 105-107 all