The Graphs of Polynomial Functions

Transcription

1 Sectio 4.3 The Graphs of Polyomial Fuctios Objective 1: Uderstadig the Defiitio of a Polyomial Fuctio Defiitio Polyomial Fuctio 1 2 The fuctio ax a 1x a 2x a1x a0 is a polyomial fuctio of degree where is a oegative iteger. The umbers a0, a1, a2,, a are called the coefficiets of the polyomial fuctio. The umber a is called the leadig coefficiet ad a0 is called the costat coefficiet , 2, ad 4 Determie if the give fuctio is a polyomial fuctio. If it is, the idetify the degree, the leadig coefficiet, ad the costat coefficiet. a) f(x) = b) f(x) = Degree Leadig Coefficiet Costat Coefficiet Degree Leadig Coefficiet Costat Coefficiet Objective 2: Sketchig the Graphs of Power Fuctios (a) x (b) 2 x (c) 3 x (d) 4 x (e) x ad 15 Use the associated power fuctio ad trasformatios to sketch the followig fuctios. a) f(x) = b) f(x) =

2 Objective 3: Determiig the Ed Behavior of Polyomial Fuctios Process for Determiig the Ed Behavior of a Polyomial Fuctio 1 2 ax a 1x a 2x a1x a0. If the degree is odd, the graph has opposite left-had ad right-had ed behavior, that is, the graph starts ad fiishes i opposite directios. Odd degree polyomials have opposite left-had ad right-had ed behavior. a 0, odd degree a 0, odd degree If the degree is eve, the graph has the same left-had ad right-had ed behavior, that is, the graph starts ad fiishes i the same directio. Eve degree polyomials have the same left-had ad right-had ed behavior. a 0, eve degree a 0, eve degree ad 21 Use the ed behavior of the graph of the give polyomial fuctio to aswer the followig: 16 a)the degree is (eve or odd). b) The leadig coefficiet is (positive or egative). 21 a)the degree is (eve or odd). b) The leadig coefficiet is (positive or egative).

3 Objective 4: Determiig the Itercepts of a Polyomial Fuctio The umber x c is called a zero of a fuctio f if f( c) 0. If c is a real umber, the c is a x- itercept. Therefore, to fid the x-itercepts of a polyomial fuctio y, we must fid the real solutios of the equatio f( x) Fid the itercepts of the polyomial fuctio f(x) =. The y-itercept is y =. The x-itercept(s) is/are x = Fid the itercepts of the polyomial fuctio f(x) =. The y-itercept is y =. The x-itercept(s) is/are x =.

4 Objective 5: Determiig the Real Zeros of Polyomial Fuctios ad Their Multiplicities The Shape of the Graph of a Polyomial Fuctio Near a Zero of Multiplicity k. Suppose c is a real zero of a polyomial fuctio f of multiplicity k, that is, x c is a factor of f. The the shape of the graph of f ear x = c is as follows: If k 1 is eve, the the graph touches the x-axis at x = c. k OR If k 1 is odd, the the graph crosses the x-axis at x = c. OR Determie the real zeros ad their multiplicities of f(x) =. a) The real zeros of the polyomial are x =. (Use a comma to separate aswers as eeded. Type a exact aswer, usig radicals as eeded.) b) The multiplicity of the zero located farthest left o the x-axis is. The multiplicity of the zero located farthest right o the x-axis is. The graph the x-axis at the leftmost zero. (touches or crosses) The graph the x-axis at the rightmost zero. (touches or crosses)

5 Objective 6: Sketchig the Graph of a Polyomial Fuctio Four-Step Process for Sketchig the Graph of a Polyomial Fuctio 1. Determie the ed behavior. 2. Plot the y-itercept f(0) a0. 3. Completely factor f to fid all real zeros ad their multiplicities*. 4. Choose a test value betwee each real zero ad sketch the graph. * This is the most difficult step ad will be discussed i further detail i the subsequet sectios of this chapter Sketch the polyomial fuctio f(x) = usig the four-step process. The left-had behavior starts ad the right-had behavior eds. The y-itercept is. The real zeros of the polyomial are x =. The multiplicity of the zero located farthest left o the x-axis is. The multiplicity of the zero located betwee the leftmost ad rightmost zeros is. The multiplicity of the zero located farthest right o the x-axis is. What is the value of the test poit at x =? y = Sketch the graph.

6 Objective 7: Determiig a Possible Equatio of a Polyomial Fuctio Give its Graph Aalyze the graph to address the followig about the polyomial fuctio it represets. a) Is the degree eve or odd? b) Is the leadig coefficiet positive or egative? c) The value of the costat coefficiet is. d) The leftmost real zero is x =, which has a multiplicity. The secod real zero from the left is x =, which has a multiplicity. The secod real zero from the right is x =, which has a multiplicity. The rightmost real zero is x =, which has a multiplicity. e) Select a possible fuctio that could be represeted by this graph.