Math Section 2.2 Polynomial Functions

Transcription

1 Math Sectio. Polyomial Fuctios Our objectives i workig with polyomial fuctios will be, first, to gather iformatio about the graph of the fuctio ad, secod, to use that iformatio to geerate a reasoably good graph without plottig a lot of poits. I later examples, we ll use iformatio give to us about the graph of a fuctio to write its equatio. A polyomial fuctio is a fuctio of the form P( x) = a a 1 x + a 1x + a x + + ax + a1x + 0 where a 0, a, a, 0 1, a are real umbers ad is a whole umber. The umber a is called the leadig coefficiet. The degree of the polyomial fuctio is. P ( 0) = a ad this umber is called the costat coefficiet. 0 Example: The graph of 3 f( x) = xx ( ) ( x + 1) is give: 1

2 From college algebra, you should be familiar with the graphs of 3 f ( x) = x ad g ( x) = x. The graph of f ( x) = x, > 0, is eve, will resemble the graph of f ( x) = x, ad the 3 graph of f ( x) = x, > 0, is odd, will resemble the graph of f ( x) = x. Next, you will eed to be able to describe the ed behavior of a fuctio. If the degree of the fuctio is eve ad a > 0, the the ed behavior of the fuctio is Degree: Eve, Coefficiet: + If the degree of the fuctio is eve ad a < 0, the the ed behavior of the fuctio is Degree: Eve, Coefficiet: - If the degree of the fuctio is odd ad a > 0, the the ed behavior of the fuctio is Degree: Odd, Coefficiet: + If the degree of the fuctio is odd ad a < 0, the the ed behavior of the fuctio is Degree: Odd, Coefficiet: - Examples:

3 Example 1: Determie the ed behavior of the fuctio: 4 a) f ( x) = x 5x + 4. b) 3 5 = 4x x + x. f ( x) x c) f ( x) = x( x 1) (x 1) (4 x). 3 3

4 Next, you should be able to fid the x itercept(s) ad the y itercept of a polyomial fuctio. You will eed to set the fuctio equal to zero ad the use the Zero Product Property to fid the x itercept(s). That meas if ab = 0, the either a = 0 or b = 0. To fid the y itercept of a fuctio, you will fid f (0). Example : Fid the x ad y itercept(s) of f ( x) = ( x 3)( x + 4)( x). I some problems, oe or more of the factors will appear more tha oce whe the fuctio is factored. The power of a factor is called its multiplicity. So 3 give P ( x) = x ( x 3) ( x + 1), the the multiplicity of the first factor is, the multiplicity of the secod factor is 3 ad the multiplicity of the third factor is 1. If the multiplicity of a factor is 1: the graph crosses the x-axis (looks like a lie there). If the multiplicity of a factor is eve: the graph touches the x-axis, but does ot cross it (looks like a parabola there). If the multiplicity of a factor is odd ad greater tha 1: the graph crosses the x-axis ad it looks like a cubic there. Example 3: Fid the x itercept(s) of the fuctio ad state the multiplicity of each. Idicate the possible behavior of the graph through each zero: 3 ( x 1) ( x + 4)( 5) f ( x) = x + 4

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6 Now we ll put all of this iformatio together to geerate the graph of a polyomial fuctio. For each problem, you ll eed to state the degree of the fuctio the leadig coefficiet of the fuctio the ed behavior of the fuctio the x ad y itercepts (ad multiplicities) behavior of the fuctio through each of the x itercepts (zeros) of the fuctio Your graph should be smooth, with o sharp corers. Note that graphs of polyomial fuctios may have peaks or valleys, but without additioal iformatio, you will ot be able to determie how high or low these poits are. Example 4: Fid the x ad y itercepts of the graph of the fuctio. State the multiplicities of the zeros of the fuctio. State the degree of the fuctio ad fid the leadig coefficiet. Idicate the ed behavior of the fuctio ad the behavior of the fuctio through each zero. Use all of this iformatio to graph the fuctio. ( )( ) ( ) 3 Px ( ) = x+ x 1 x 4. 5

7 Example 5: Write the equatio of the cubic polyomial P (x) with leadig coefficiet - whose graph passes through (, 16) ad is taget to the x-axis at the origi. Example 6: Write the equatio of a 5 th degree polyomial with leadig coefficiet -1 give that the graph of the polyomial is taget to the x-axis at the poits ad 4 ad the graph passes through the origi. 6

8 Example 7: Give the graph of a polyomial, try to determie the equatio of the polyomial. We may ot have eough time to solve all the examples here. Sice graphig polyomials is a subject covered i College Algebra, we assume that you are already 7

9 familiar with this subject. If you are ot comfortable with graphig polyomials, please study! You ca check Chapter 4 of the olie textbook for College Algebra (the lik is o your CASA accout!). Please solve these extra problems to practice. (Extra) Example: Fid the x ad y itercepts of the graph of the fuctio. State the multiplicities of the zeros of the fuctio. State the degree of the fuctio ad fid the leadig coefficiet. Idicate the ed behavior of the fuctio ad the behavior of the fuctio through each zero. Use all of this iformatio to graph the fuctio. ( ) ( ) ( ) Px ( ) = x+ 3 x 1 x. (Extra) Example: Fid the x ad y itercepts of the graph of the fuctio. State the multiplicities of the zeros of the fuctio. State the degree of the fuctio ad fid the leadig coefficiet. Idicate the ed behavior of the fuctio ad the behavior of the fuctio through each zero. Use all of this iformatio to graph the fuctio. 3 P ( x) = x 4x x + 4. (Factor first!!!) 8

10 (Extra) Example: Fid the x ad y itercepts of the graph of the fuctio. State the multiplicities of the zeros of the fuctio. State the degree of the fuctio ad fid the leadig coefficiet. Idicate the ed behavior of the fuctio ad the behavior of the fuctio through each zero. Use all of this iformatio to graph the fuctio. ( 4 x)( x 1) ( 5) 3 P ( x) = x +. NOTE: With some problems, you ca use trasformatios to graph polyomial fuctios. (Extra) Example: Graph usig trasformatios: f ( x) = ( x 1) 3 4 9