Precalculus MATH 1730
pellissippi state community college
Popular in Course
Popular in Mathematics (M)
This 0 page Class Notes was uploaded by Brown Lowe on Sunday November 1, 2015. The Class Notes belongs to MATH 1730 at pellissippi state community college taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/232970/math-1730-pellissippi-state-community-college in Mathematics (M) at pellissippi state community college.
Reviews for Precalculus
Report this Material
What is Karma?
Karma is the currency of StudySoup.
Date Created: 11/01/15
ZEROS 0F POLYNOMIAL FUNCTIONS Summary of Properties 1 The function given by fx anx n an1xquot 1 xn2xquot 2 a2x2 a1x an is called a polynomial function ofx with degree n where n is a nonnegative integer and an an1 an2 a2 a1 an are real numbers with an 0 2 The graphs of polynomial functions are continuous and have no sharp corners The sign of the leading coef cient an determines the end behavior of the function The degree n determines the number of complex zeros ofthe function The number of real zeros ofthe function will be less than or equal tothe number of complex zeros 3 The real zeros of a polynomial function may be found by factoring where possible or by nding where the graph touches the xaxis The number of times a zero occurs is called its multiplicity If a function has a zero of odd multiplicity the graph ofthe function crosses the xaxis at that xvalue However if a function has a zero of even multiplicity the graph of the function only touches the xaxis at that xvalue 4 The graphing calculator has a builtin function for nding a zero or root of a function As an alternative method you can graph y 0 the xaxis as a second function and use the intersection function to nd the zero While this latter method is somewhat easier to use on some calculators it may not work for finding zeros of even multiplicity Finding the Zeros of Polynomial Functions Find the real zeros and state the multiplicity of each for the following polynomial functions Algebraic solution Graphical solution 1 fx4x2 3x 7 4x2 3x 70 4x 7x10 4 x 70 or x10 X1 or x 1 E0 4 x17s 0 Each zero has multiplicity one Repeat to nd other zero algebraic solution graphical solution 2 fxx41 x410 x4 1 has no real solutions This function has no real zeros algebraic solution 3 fx x72x5 x3 x7 2x5 x30 x3x4 2x210 x3x212 X3X 1x1X 1x10 x 0 or x 120 or x120 x0 or x1 or x1 graphical solution l The zeros of the function are 0 multiplicity 3 1 multiplicity 2 and 1 multiplicity 1 Writing Polynomial Functions with Specified Zeros Write an equation of a polynomial function of degree 3 which has zeros of 0 2 and 5 General solution Any function oftheform fx axx 2x 5 where a 0 will have the required zeros Speci c solutions fx xx 2x 5 x3 3x2 10x gx 3xx 2x 5 3x3 9x2 30x Write an equation of a polynomial function of degree 7 which has zeros of 0 multiplicity 2 2 multiplicity 3 and 5 multiplicity 2 General solution Any function oftheform fx ax2x 23x 52 where a 0 will have the required zeros Write an equation ofa polynomial function of degree 2 which has zero 4 multiplicity 2 d and opens downwar Atypical solution is fx 3x 42 The leading coef cient must be negative Write an equation of a polynomial function of degree 3 which has zeros of 2 2 and 6 and which passes through the point 3 4 Solution fx ax 2x 2x 6 hasthe required zeros f3a323 23 64 15a4 a fX X 2X 2X 6 has the required zeros and passes through the specified point Exercises A Algebraically find the exact real zeros and state the multiplicity of each 1 fX 2x2 9x 5 2 fX 9x2 24x 16 3 fX 9x2 4 4 fX 9x2 4 5 fX 2x2 4x 1 6 fX 8x3 27 7 fX 3x5 5x4 x3 8 fX 32x3 4 9 fX 1 x4 10 fX 2x4 26x2 72 11 fX 4x4 36 12 fX x32x 13x4 16 B Graphically find the real zeros and state the multiplicity of each Round answers to 4 decimal places 13 fXX2X 1 14 fXX3 3X22X 4 15 fX x3 4x22x1 RATIONAL FUNCTIONS AND THEIR GRAPHS l fis a rational function if fX where gX and hX are polynomials hX 0 X HINT Look for a variable in the denominator The domain of a rational function consists ofall real numbers excth the zeros of the denominator To nd the zeros ofthe denominator set it equal to 0 and solve for X A fX2X2 X X 6 1 X2 X 60 X 3X20 x3 2 2 Domainoffis 2 23 3 o B lfthe denominator does not equal 0 for any real number then the domain is Ill The line X a is a vertical asymptote for the graph of fX if fX or fX as X afrom either side This means that as X gets closer and closer to a the graph of fX gets closer and closer to the line X a without touching it A In the example above the vertical asymptotes are X 3 and X 2 B Although there is not a limit to the number of vertical asymptotes a rational function may have it is also possible that a rational function may have no vertical asymptote IV The equations of the vertical asymptotes are X a X b where a b are zeros ofthe denominator lfthe denominator cannot equal 0 then there are no vertical asymptotes lfa factor of the denominator cancels out there will be no vertical asymptote at that zero 2 A fX i has no vertical asymptotes since X2 1 o O X 1 B f00MW X23x2 X1x2 1 fhas one vertical asymptote X 1 2 Since X 2 cancels out there is no asymptote at X 2 3 However there is a hole in the graph at X 2 2 is not in domain of fX C The graph can nevertouch a vertical asymptote V The line y c is a horizontal asymptote forthe graph of fX if fX c as X o or fX c as X o This means that as X gets very large positively or negatively some portion of the graph of fX approaches the line y c without touching it However some other portion ofthe graph may touch or even cross the line y c Vl Since the horizontal asymptote is determined by the behavior ofthe function as X becomes very large we consider only the terms of highest degree in the numerator and denominator We will consider three cases 3 A Degree of numerator is less than degree of denominator fX 4 X 1 As X gets larger and larger without bound the y fX gets smaller and smaller 2 The horizontal asymptote is y 0 3x2 8 B Degree of numerator equals degree of denomInator y 2 X 4X 1 As X gets larger and larger without bound the dominating terms are those with the highest degrees 3X2 2AsXy 3 X2 3 The horizontal asymptote is y 3 3 2 C Degree ofnumerator is one larger than degree of denominator fX X2X 6 X 4 1 As X becomes large without bound the numerator increases faster than the the denominator 2 There is no horizontal asymptote X 1 3 X2 4 X3 X20X 6 fXX 1 I3 4x X2 4 X24X 6 x2 4 4X 10 4 The oblique or slant asymptote is y X 1 VII The range ofa rational function can be dif cult to determine It is frequently helpful to consider both the equation and the graph A The range of a rational function with linear numerators and denominators can be readily determined 4 10X B X g 5X 1 is 2 2 has a horizontal asymptote ofy 2 which implies that its range 3x2 3x e VIII The domain of fx is 4 4 and its horizontal asymptote x2 ax 16 is y 3 A Can fx 3 2 B fx m 3 implies 3x2 3x e 3x2 ax 16 Solving this 2 BX 16 equation shows that x 2 Note that 2 is in the domain ofthe function 2 nranh us y 3 at the point 2 3 IX Sketching the graph of a rational function fx 6x2x 3x 3 4 x2x 2 x 2 IF x 2 There is a hole at 2 f 2 2 Plot this point as an open circle B The vertical asymptote is x 2 Plot this as a dashed line C The domain is 2 2 2 2 D The horizontal asymptote is y 1 Plot this as a dashed line 2 E Although X seems to imply that x 2 we must rememberthat 2 39 quot quot 39 the function Therefnrs horizontal asymptote F The range is 1 1 G Letting x 0 shows that 0 15 is the yintercept Plot this point H Letting y 0 shows that 3 0 is the xintercept Plot this point I r Make use ofthe end behavior as you drawthe graph