Constant no variables, just a number. Linear Note: Same form as f () x mx b. Quadratic Note: Same form as. Cubic x to the third power

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1 Precalculus Notes: Section. Modeling High Degree Polnomial Functions Graphs of Polnomials Polnomial Notation f ( ) a a a... a a a is a polnomial function of degree n. n n 1 n n n1 n 1 0 n is the degree of the polnomial because it is the highest power of in the polnomial. When n is used as a subscript to a it is just a means of numbering the coefficients. An eample of a polnomial of degree 4 is 4 f ( ) 5 6. What is a in this polnomial? What is a1 in this polnomial? 0 Common Polnomial Names Know These Names f () a Constant no variables, just a number 0 f () a a Linear Note: Same form as f () m b 1 0 f () a a a Quadratic Note: Same form as 1 0 f () a b c f () a a a a Cubic to the third power 1 0 f () a a a a a Quartic to the fourth power

2 Graphs of Common Polnomials and Their End Behaviors (The limits are their end behaviors.) Quadratic f ( ) 1 As gets infinitel large, gets infinitel small. As gets infinitel small, gets infinitel small. f ( ) Cubic f () f () f ()

3 Quartic 4 f () 4 f () f () 4 4 f () For polnomials with an even degree, the left and right end behaviors are the same. For polnomials with an odd degree, the left and right end behaviors are opposites. Finding the Zeros of a Polnomial Function b Factoring (1. Set the polnomial equal to zero,. factor it, then. set each factor equal to zero.) Find the zeros of f ( ) 4 4. Find the zeros of f ( ) f 0 ( ) 4 4 0, ( ) f 0,,

4 Finding the Zeros and Etrema of a Polnomial Function Using Graphing Technolog Graph the given function in an appropriate viewing window, then find all its -intercepts and etreme values. Look in the CALC ( nd TRACE) menu to find the zeros, maimum and minimum values. f ( ) 7 -intercepts 1.76,0,.59,0 & 4.467,0 local maimum.86,19.17 local minimum.86, 5.17 f 4 ( ) 7 8 -intercepts 4.5,0 &,0 local maimum.1, local minima.7, & 1.155, 1.46

5 Theorem : Local etrema and zeros of Polnomial Functions (Page 0 green bo) A polnomial function of degree n has at most n 1 local etrema and at most n zeros Definition: Multiplicit of a zero of a polnomial function (Page 05 green bo) If f is a polnomial function and of f. m m 1 is a factor of f but c c E. = ( 4) ( ) ( 1) has zeros: = 4 with multiplicit, = - with multiplicit, and = 1 with multiplicit 1. is not, then c is a zero of multiplicit m Zeros of odd and even multiplicit (Page 06 green bo) If a polnomial function f has a real zero c of odd multiplicit, then the graph of f crosses the -ais at (c, 0), and the value of f changes sign at = c. Laman s terms : At a zero of odd multiplicit, the graph crosses through the ais. If the odd multiplicit is greater than 1 (e:, 5, 7, etc.) then there is a flattening effect that takes place as the graph crosses through the ais. If a polnomial function f has a real zero c of even multiplicit, then the graph of f does not cross the - ais at (c, 0), and the value of f does not change sign at = c. Laman s terms : At a zero of even multiplicit, the graph has a local etrema (local ma or min) tangent (just touching) to the -ais, but the graph does not cross through the -ais. Intermediate value Theorem (Page 06 green bo) If a and b are real numbers with a < b and if f is continuous on the interval [a, b], then f takes on ever value between f(a) and f(b). In other words, if 0 is between f(a) and f(b), then 0 fc () for some number c in [a, b]. In particular, if f(a) and f(b) have opposite signs (i.e., one is negative and the other is positive), then f(c) = 0 for some number c in [a,b], Laman s terms : If f is continuous and the sign of f goes from negative to positive, there must be an intercept (zero) between the values for the negative and positive outputs for f.

6 For the given polnomial, sketch a graph and determine the following: f ( ) ( 1)( )( ) Tpe (name) of polnomial Leading Term CUBIC List and label the -intercepts (zeros, roots) and their multilplicities. = - M1, =-1 M1, = M1 Left End Behavior Right End Behavior -intercept (0,-6),0 1,0,0 f ( ) ( ) ( ) 1 Tpe (name) of polnomial QUARTIC,0,0 Leading term 4 List and label the -intercepts (zeros, roots) and their multilplicities. = -1 M1, = M, = M1 Left End Behavior Right End Behavior -intercept (0,-1) 1,0

7 f ( ) ( ) 1 Tpe (name) of polnomial Leading term 4 QUARTIC List and label the -intercepts (zeros, roots) and their multilplicities. = -1 M1, = M Left End Behavior Right End Behavior -intercept (0,-8) 1,0,0 Using the data points in the table shown, use a cubic regression to determine a cubic polnomial that passes through the points STAT EDIT Enter -values in L1, -values in L STAT CALC CubicReg f ( )

8 Using onl algebra, find a cubic function that has the given zeros. 1, 1, 5 f ( ) f ( ) f ( ) Using onl algebra, find a cubic function that has the given zeros and also passes through (1,).,, f ( ) a f ( ) a 4 f a ( ) 4 1 a(1 41) a(6) 1 a 1 f ( ) 4 1