Analysis of Polynomial & Rational Functions ( summary )
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1 Analysis of Polynoial & Rational Functions ( suary ) The standard for of a polynoial function is ( ) where each of the nubers are called the coefficients. The polynoial of is said to have degree n, where n The lead coefficient is, the constant ter is, the doinating ter is Note: a degree 0 polynoial is siply a constant, f(x)= (1) In general we can allow the coefficients to be real ( ) or iaginary ( ) ( ) ( ) but in ost of our probles they will be real and in ost probles they will also be nice integers! (i) If we divide a polynoial f(x) by another g(x) we get have a quotient ( ) ( ), the nuerator is called the dividend and the denoinator, the divisor. The division will result in a quotient polynoial q(x) and a reainder polynoial r(x) where f,g,q,r are related by ( ) ( ) ( ) ( ). The degree of the reainder polynoial will be such that 0 ( ) ( ) i.e. The Division Algorith. *As a special case if the divisor g(x) is a binoial of the for x c then ( ) ( )( ) where the reainder r is a siple constant *The division f(x) by x c can by done using synthetic division * When f(x) is divided by the binoial x c then r = f(c) i.e. The Reainder Theore (ii) We can factor a polynoial by knowing its zeros: The binoial x c is a factor of f(x) iff f(c)=0 i.e. The Factor theore () The zero s of f ay be real, iaginary, or purely iaginary ( or a cobination of all three ). (a) If we would like to know how any of the there are i.e. Nuber of Zero s theore (b) If we would like to know how any are positive real, how any are negative real, and how any are iaginary i.e. Descarte s Rule of Signs ( this gives us a table of ultiple possibilities ) (c) If they are rational zero s, are they liited to a certain group of fractions? i.e. Rational Zero s Theore ( this allows us to construct a list of rational nubers that are possible zeros to f ) (d) If we would like to know if the zero s are bounded, naely the zero s will lie soewhere between a negative lower bound and positive upper bound ( analogous to a house that lies in given area of land bounded by its fence ) i.e. The Bounds Theore () The zero s and their relation to the graph of the polynoial 1
2 (a) A real zero of f shows up as an x-intercept and the nuber of x-intercepts will not exceed the degree of the polynoial ( i.e. A degree n polynoial can never have ore than n x intercepts ) (b) The local behavior of the graph near a zero ( passes straight through, wiggles, bounces ) depends on the ultiplicity of the zero (i) if C a zero of ultiplicity 1 (ii) if C is a zero of odd ultiplicity C (iii) if C is a zero of even ultiplicity C C (c) Between any two successive zero s there s at least one turning point (d) If the coefficients of the polynoial function are real then if z is an iaginary zero of f, its coplex conjugate ust also be a zero i.e. coplex conjugate zeros theore (4) The nuber of turning points & the range of a polynoial function (i) The nuber of turning points will be n 1 or less ( i.e. the nuber of turning points is at ost n 1) (ii) If n is odd then Range = entire real line = ( - ) (iii) If n is even then Range will be of the for ( - or [ k, ) for soe constant k (5) A polynoial function has four possible end behaviors Right x, Right x, Right x, Right x,
3 ( 6 ) The relationship between factoring a polynoial function and finding its zeros Three cases outlined below Start (I) You are given the factored for of f(x) End Get zeros of f(x) and their ultiplicities by siple observation of the factored for zero ultiplicity f ( x) 7( x ) ( x 5) -5 (II) The lead coefficient A for f(x) is known as well as its zeros and ult You can f ( x) A x n a x n 1... a x a n easily factor f(x) f ( x) A ( x b) k ( x c) l ( x d) zero b c d ultiplicity k l f ( x) 6x 4 7x -x 15 Has zeros - /, -/, 1-i, 1+i f ( x) 6( x ) ( x [1 i])( x [1 i])
4 (III) Only the zeros are known but you also know a point (x,y)=(a,b) on the graph of f zero ultiplicity You can factor f(x) easily but you ust use the point (a,b) given to deterine the value of the lead coefficient A f ( x) A ( x b) k ( x c) l ( x d) b c d k l f has zeros 1, - and f(0)= f(x)= A(x-1)(x+) f(0)=-6 eans A = So f(x) = - (x 1)( x + ) (7) How to graph a polynoial function f Step (1) Factor f(x) by finding the zeros and their ultiplicites f ( x) A ( x b) k ( x c) l ( x d) Step () Plot the zeros and graph the local behavior of the function near the zeros Step () Plot the x and y intercepts Step () Make a table of (x,y) values for several values of x that you choose Step (4) Deterine the End Behavior of the graph and see that it coincides with your plotted points 4
5 Rational Functions A rational function is the ratio of two polynoials p, q px ( ) f( x) qx ( ) In our probles p, q will have no coon factors. That is, p/q has be reduced to lowest ters Note: if p, q do have coon factors then this introduces a hole in the graph at the value that akes the denoinator zero i.e. x f( x) x 4 8 ( x )( x ) can be written f( x) ( x )( x x 4) and hence has coon factor (x-) and hence a hole in the graph at x = x Once you cancel like factors f ( x) provided x x x 4 and a hole at 1 (, ) (1) Find equations of vertical asyptotes: Any x that satisfies q(x)=0 Definition : The line x = c is a vertical asyptote of the function f provided As x c, f ( x) NOTE: this is equivalent o the books definition of, As x c, f ( x) This leads to one of 4 possible local asyptotic behaviors at near x = c (i) (ii) (iii) (iv) Note: to deterine which case we siply chose an x value close to c but to the right and left of c and find the sign of f(x) whether + or - 5
6 () To find the equations of horizontal aysptotes we use the Horizontal Asyptotes Theore Definition: the line y = c is a horizontal asyptote of the function f provided As x, f ( x) c NOTE: this is equivalent to the books definition of, As x, f ( x) c The horizontal asyptote theore leads to one 4 possible local asyptotic behaviors at y = c (i) (ii) (iii) (iv) NOTE(a) If the asyptote is the x-axis, naely y = 0, then you need only let x be a large positive nuber ( like 1000) and large negative nuber ( like ) and check for the sign of f(x) to deterine which of the 4 cases above applies (b) If the asyptote isn t the x-axis, naely y = c, c not 0, then you need to know the nuerical value of f(x) when x is a large positive nuber and large negative nuber to deterine which of the 4 cases above applies ( ) Oblique Asyptotes: Suppose that px ( ) f ( x), degree p( x) n degree q( x) k qx ( ) Oblique asyptotes y = x + b occur in the special case that n = k + 1 To find the equation y = x +b first divide p(x) by q(x) using long division and the x + b part is the quotient in the long division 6
7 (4) Can a rational function intercept its horizontal asyptote? Yes, it ay. So you have to check for this. Suppose you know the horizontal asyptote is y = c, then you let f(x) = c so you solve the equation px ( ) c qx ( ) If this has a solution then this is the x-value of the point (x,y) of intersection. If this equation does not have a solution, then there is no intersection. (5) Graphing a rational function: Step (i) Find vertical asyptotes and graph with a dotted line, find horizontal or oblique asyptotes ( if any ) and graph with a dotted line Step (ii) Deterine the local behavior near all the asyptotes Step (iii) Check for any point of intersection (x,y) of the function with its horizontal asyptote Step (iv) Plot the x and y intercepts Step (v) Make a table of (x,y) values for several values of x that you choose 7
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