Polynomial Functions. New Section 1 Page 1. A Polynomial function of degree n is written is the form:
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1 New Sectio 1 Page 1 A Polyomial fuctio of degree is writte is the form: 1 P x a x a x a x a x a x a where is a o-egative iteger expoet ad a 0 ca oly take o values where a, a, a,..., a, a a. a 0 Example: For each of the followig, determie if the give fuctio is a polyomial. If it is a polyomial state the degree ad the leadig coefficiet. 4 f x 8x x 1x 4 g x x 4 x 10x 6 4 h x x x 8 x 5 j x x x Graphig a polyomial fuctio is a ecessity i this course ad all future courses. I calculus, you will lear that the graph of a polyomial fuctio is always a smooth curve (o sharp corers or cusps) ad cotiuous (o holes or gaps).
2 New Sectio 1 Page Let's ivestigate the graph of the powers fuctios: f x Ax where is ay o-egative iteger. Take the first few EVEN powers of, (These are EVEN fuctios) 4 6,, f x x x x Behavior of the fuctio about the x-itercept/real zero: Ed Behavior of the fuctio: (Basically, describe what y is doig as x grows without boud) What if the leadig coefficiet is NEGATIVE? (how will this affect the ed behavior) Take the first few ODD powers of, (These are ODD fuctios) 5 7,, g x x x x Behavior of the fuctio about the x-itercept/real zero: Ed Behavior of the fuctio: (Basically, describe what y is doig as x grows without boud) What if the leadig coefficiet is NEGATIVE? (how will this affect the ed behavior)
3 New Sectio 1 Page Sketch the fuctio, f x x based o the trasformatio of x Sketch the fuctio, x 6 g x based o the trasformatio of 6 x SUMMARIZE THE END BEHAVIOR OF A POLYNOMIAL FUNCTION A th degree polyomial fuctio is oly as good as its HIGHEST DEGREE TERM. That is, look at the degree of the polyomial ad the leadig coefficiet to determie the behavior of the polyomial as x gets large. That is, P x a x a x a x a x a x a behaves as the moomial, y a x EVEN DEGREE ODD DEGREE
4 New Sectio 1 Page 4 The Behavior of a Polyomial Fuctio Near a x-itercept/real zero. The x-itercepts/real zeroes play a major role i the graph of a polyomial fuctio. At the x-itercepts/real zeroes the graph must either cross the x-axis or touch the x-axis. Also, i betwee cosecutive real zeroes the graph will either be above the x-axis or below the x-axis. Example: Cosider the followig fuctio is factored form: f x x x y-itercept: Real Zero's: Set f(x) = 0 The fuctio above is a perfect example of a rd degree polyomial with a positive leadig coefficiet. The real zero of x = - is said to have multiplicity of 1 while the real zero of x = is said to have a multiplicity of. Multiplicity refers to the umber of times the zero occurs. Example: Fid a possible 4th degree polyomial whose real zero's are with a multiplicity of, ad ad - both with a multiplicity of 1.
5 New Sectio 1 Page 5 Let f(x) be a polyomial fuctio ad suppose that x a is a factor of f(x). [Furthermore, assume that oe of the other factors of f(x) cotais (x-a).] The, i the immediate viciity of the x-itercept/real zero at a, (here a is called a zero of multiplicity ) the the graph of y = f(x) closely resembles that of y A xa The priciple that we have just stated is easy to apply because we already kow how to graph fuctios of the form y A xa. The ext example shows how this works. f x x x x Real Zero's: f x x x x Real Zero's:
6 New Sectio 1 Page 6 A Approach to Graphig 1. Factor the polyomial, if possible.. Fid the y-itercept, set x = 0.. Fid the real zeroes, set y = 0 ad solve for x. 4. Determie the behavior ear the real zero. Eve Multiplicity graph touches (bouces) at the real zero. -- Sig value of f(x) does NOT chage o either side of the real zero. Odd Multiplicity graph crosses (cuts) at the real zero. -- Sig value of f(x) CHANGES sig from oe side of the real zero to the other. 5. Determie the ed behavior. Look at the degree of the polyomial ad the leadig coefficiet to determie the behavior as x gets large. 6. Plot poits as ecessary to determie the geeral shape of the graph. But, all we really eed is oe poit i ay iterval to determie whether the graph is above or below the x -axis i that iterval, the we ca fill i the rest based o the kowledge of the multiplicity of the real zero. Example: Sketch the fuctio based o the above iformatio: f x x x Example: Sketch the fuctio based o the above iformatio: g x x x x
7 New Sectio 1 Page 7 Fu Fact: A polyomial fuctio of degree ca have AT MOST -1 turig poits Cosider the third degree polyomial: f x x 4 g x x x x Example: Cosider the graph of the followig polyomial fuctio f(x). Assume the polyomial has o turig poits beyod those show. Is the leadig coefficiet POSITIVE or NEGATIVE? What is the miimum degree of this polyomial? Fid ad describe the real zeros (ad otice the multiplicity) Fid a possible polyomial that could model this graph if factored form.
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