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Class Note for MATH 1330 at UH 6

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This 7 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 15 views.

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Date Created: 02/06/15
Math 1330 Section 22 Polynomial Functions Our objectives in working with polynomial functions will be rst to gather information about the graph of the lnction and second to use that information to generate a reasonably good graph without plotting a lot of points In later examples we ll use information given to us about the graph of a function to write its equation A polynomial function is a function of the form Px anxquot aHx39H anizx39 Z 6sz2 a1x do where an 0 a0 a1 an are real numbers and n is a whole number The number an is called the leading coef cient The degree of the polynomial lnction is n PO a0 From college algebra you should be familiar with the graphs of f x x2 and gx x3 The graph of f x xquot n gt 0 n is even will resemble the graph of f x x2 and the graph of fx xquot n gt 0 n is odd will resemble the graph of fx x3 19st 59 797397575473441 3456 agi NeXt you will need to be able to describe the end behavior of a function If the degree of the function is even and an gt 0 then the end behavior of the function is If the degree of the function is even and an lt 0 then the end behavior of the function is If the degree of the function is odd and an lt 0 then the end behavior of the function is NeXt you should be able to nd the x intercepts and the y intercept of a polynomial function You will need to set the function equal to zero and then use the Zero Product Property to nd the x intercepts That means if ab 0 then either a 0 or b 0 To nd the y intercept of a function you will nd f 0 In some problems one or more of the factors will appear more than once when the function is factored The power of a factor is called its multiplicity So givenPx x2 x 33 x 1 then the multiplicity of the rst factor is 2 the multiplicity of the second factor is 3 and the multiplicity of the third factor is 1 If the multiplicity of a factor is 1 then the graph of the function will resemble y x at the zero generated by the factor If the multiplicity of a factor is even then the graph of the function will resemble y x2 at the zero generated by the factor If the multiplicity of a factor is greater than 1 and is odd then the graph of the function will resemble y x3 at the zero generated by the factor You can use all of this information to generate the graph of a polynomial function degree of the function end behavior of the function x and y intercepts and multiplicities behavior of the function through each of the x intercepts zeros of the function Graphs of polynomial functions may have peaks or valleys but Without additional information you will not be able to determine how high or low these points are Example 1 Find thex andy intercepts ofthe graph ofthe function State the degree ofthe function Px x 7 3Xx 1x 2 Example 2 Find thex andy intercepts ofthe graph ofthe function State the degree ofthe function Px x3 7 x2 7x 1 Example 3 State the x andy intercepm ofthe graph ofthe function Px xx 7 3Xx 1 State the degree of the function Then sketch the graph of the function labeling all intercepts Show the correct behavior through each x intercept and show the proper end behavior Example 4 State the x and y intercepts of the graph of the function Px 3 7 xXx lx 5Z State the degree of the function Then sketch the graph of the function labeling all intercepts Show the correct behavior through each x intercept and show the proper end behavior Example 5 State the x and y intercepts of the graph of the function Px 4x lzx 7 2Zl 7 x State the degree ofthe function Then sketch the graph ofthe function labeling all intercepts Show the correct behavior through each x intercept and show the proper end behavior Example 6 State the x andy intercepts ofthe graph ofthe function Px x 53x 7 2Xx712 State the degree of the function Then sketch the graph of the function labeling all intercepts Show the correct behavior through each x intercept and shoW the proper end behavior 79737 if 75473 771 1 Example 7 Write the equation of the cubic polynomial Px that satis es the following conditions zeros at x 3 x 71 and x 4 and passes through the point 3 7 Example 8 Write the equation of the quartic function With y intercept 4 Which is tangent to the x axis at the points 1 0 and l 0 With some problems you can use transformations to graph polynomial functions Example 9 Gmph using tmnsformations f x x372 79 Example 10 Graph using transformations f x 7x l4 3 a iyiai issi il il 1 Axtrsax

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