2) Give an example of a polynomial function of degree 4 with leading coefficient of -6

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1 Math 165 Read ahead some cocepts from sectios 4.1 Read the book or the power poit presetatios for this sectio to complete pages 1 ad 2 Please, do ot complete the other pages of the hadout If you wat to move ahead, solve problems from MXL Sectio 4.1 Polyomial fuctios 1) The polyomial fuctio is of the form: 2) Give a example of a polyomial fuctio of degree 4 with leadig coefficiet of -6 3) Give a example of a o-polyomial fuctio. 4) Is this the graph of a polyomial fuctio? Explai why or why ot. Sectio 4.1 Power Fuctios Graphig Power Fuctios of the form Px ( ) = x - Eve degree 5) Use calculator to graph each of the followig fuctios. 2 4 f( x) = x gx ( ) = x hx ( ) = x 6 6) Commet o each of the followig properties of power fuctios of eve degree : Symmetry Domai Rage Cotais what poits Steepess 1

2 Sectio 4.1 Power Fuctios Ed Behavior - Trasformatios Graphig Power Fuctios of the form Px ( ) = x - Odd degree 7) Use calculator to graph each of the followig fuctios. 3 5 f( x) = x gx ( ) = x hx ( ) = x 7 8) Commet o each of the followig properties of power fuctios of odd degree : Symmetry Domai Rage Cotais what poits Steepess Ed Behavior Summary: P( x) = ax (We will do this summary i class) 9) is eve with a > 0 ad a < 0 10) is odd with a > 0 ad a < 0 11) Graphig Trasformatios of Power Fuctios review sectio 2.5 I each case, idicate the trasformatios that took place. 3 a) Graph f( x) = ( x 2) b) Graph Px 4 ( ) = 2( x+ 4) 8 c) Graph hx 5 ( ) 3x 9 = d) Graph 1 hx = x ( ) 15 2

3 Sectio 4.1 Polyomial fuctios Ed Behavior Importace of Leadig Term i Polyomial Fuctios aa xx 12) Sketch the graphs of the give fuctios Sketch the graph (o calculator eeded) Sketch the graph of these fuctios (use calculator, if ecessary) ff(xx) = 2xx 2 ff(xx) = 2xx 2 + 3xx 5 gg(xx) = xx 6 gg(xx) = xx 6 9xx 2 h(xx) = 2xx 3 h(xx) = 2xx 3 10xx kk(xx) = xx 4 kk(xx) = xx 4 + 5xx ) Summarizig Ed Behavior of Polyomial Fuctios ff(xx) = aa xx 3

4 Sectio 4.1 Zeros ad Graphs of Polyomial fuctios 14) Fid the zeros. Use the zeros ad the ed behavior to graph each fuctio. (do ot use the calculator) a) ff(xx) = xx 2 + 6xx 7 Zeros: What is the leadig term? EB with arrows Graph b) ff(xx) = 3(xx + 1)(xx 5)(xx 7) Zeros: What is the leadig term? EB with arrows Graph c) ff(xx) = 2(xx + 5)(xx + 2)(xx 1)(xx 4) Zeros: What is the leadig term? EB with arrows Graph 15) Summarizig Zeros ad Factors - Complete the followig: 16) Complete the followig equivalet statemets: a) r is a of the polyomial fuctio f(x) b) r is a of the graph of f c) is a factor of f d) r is a solutio to the equatio 4

5 Sectio 4.1 Polyomial fuctios Multiplicity of the zeros 17) Idetify the leadig term, the ed behavior, the zeros, ad sketch the graphs (o calculator) a) ff(xx) = 2 (xx 1) LT = EB = b) kk(xx) = 0.5(xx + 2) 2 LT = EB = c) h(xx) = 3(xx 5) 3 LT = EB = d) gg(xx) = 0.01(xx + 2) 2 (xx 1)(xx 5) 3 LT = EB = 18) Summarizig Multiplicity - Complete the followig: the, r is called a 1) If r is a zero of eve multiplicity, the the graph the x axis at r. 2) If r is a zero of odd multiplicity, the the graph the x axis at r. Note: Multiplicity 1 Odd Multiplicity higher tha 1 5

6 Sectio 4.1 Polyomial fuctios Turig poits 19) Summarizig Turig Poits - For each of the polyomial fuctios give o previous pages, give the degree ad cout the umber of turig poits. The umber of turig poits of a polyomial fuctio of degree is at most Graphig Polyomial Fuctios i Factored form 20) Sketch polyomial fuctios give i factored form. Discuss leadig term, degree, umber turig poits, zeros, multiplicities, behavior at the x-axis, ad ed behavior. a) gg(xx) = 0.5(xx + 2) 2 (xx 1) 3 (xx 5) 3 LT = Degree = EB = Multiplicities = Behavior at the x-axis = Graph Check TP = b) gg(xx) = 1.5(xx + 2) 3 (xx 1) 2 (xx 4)(xx 8) LT = Degree = EB = Multiplicities = Behavior at the x-axis = Graph Check TP = 6

7 Sectio 4.1 Writig Polyomial Fuctios with give characteristics 21) Write a polyomial fuctio with give degree, x-itercepts ad a specific y-itercept (or ay other poit). Sketch graph; the, check with calculator. a) Degree 5, zeros: -1 (M2), 5(M3), through (0,-2) b) Degree 6, zeros: -3(M2), 1(M1), 3(M3), through (5,-7) 7

8 Sectio 4.1 Polyomial fuctios From a graph to a fuctio 22) Complete the followig For aa idicate positive or egative. For the degree, idicate eve or odd a) Polyomial goes through (7, 2) (b) Polyomial goes through (0, 9) (c) y-itercept is (0, -5) (d) Poit o graph (0, -1/3) 8

9 Sectio 4.1 Polyomial fuctios From a graph to a fuctio 23) Comprehesive graphs of polyomial fuctios are give. Complete the followig. This time, the leadig coefficiet is either aa = 1 oooo aa = 1 (a) (b) (c) (d) 24) Aswer the followig: 9

10 Sectio Summarizig Characteristics of Polyomial Fuctios Turig poits ad zeros 1) A 4 th degree polyomial has at most turig poits, ad at most real zeros. 2) A th degree polyomial fuctio has at most turig poits, ad at most real zeros. 3) The miimum umber of real zeros of a odd degree polyomial fuctio is 4) The miimum umber of real zeros of a eve degree polyomial fuctio is Ed behavior 5) For ay degree, if 0 a >, the right ed behavior is UP / DOWN 6) For ay degree, if a < 0, the right ed behavior is UP / DOWN 7) Summary for ed behavior Idicate with arrows the possibilities for the ed behavior of Eve degree polyomial fuctio Odd degree polyomial fuctio a > 0 a < 0 Rage 6) If is odd, the rage of the polyomial fuctio of degree is 7) Idicate possibilities for the rage of eve degree polyomial fuctios. Summary for rage 8) Write the possibilities for the rage based o the degree of the polyomial Rage of a eve degree polyomial fuctio Rage of a odd degree polyomial fuctio a > 0 a < 0 Zeros, factors, multiplicity ad behavior at the x-coordiates of the zeros 9) If a polyomial fuctio has a zero of multiplicity 1 at x = b, the the fuctio has a factor ad the graph the x axes at x = 10) If a polyomial fuctio has a zero of odd multiplicity M (M > 1) at x = c, the the fuctio has a factor ad the graph at x = 11) If a polyomial fuctio has a zero of eve multiplicity K at x = d, the the fuctio has a factor ad the graph at x = 10