Polyomial Fuctios ad Their Graphs I this sectio we begi the study of fuctios defied by polyomial expressios. Polyomial ad ratioal fuctios are the most commo fuctios used to model data, ad are used extesively i mathematical models of productio costs, cosumer demads, wildlife maagemet, biological processes, ad may other scietific studies. Usig these fuctios ad their graphs, predictios regardig future treds ca be made. Polyomial Fuctios ad Their Graphs: Before we start lookig at polyomials, we should kow some commo termiology. Defiitio: A polyomial of degree is a fuctio of the form 1 1... P x = a x + a x + + a x + a 1 1 0 where a 0. The umbers a0, a1, a,..., a are called the coefficiets of the polyomial. The umber a 0 is the costat coefficiet or costat term. The umber a, the coefficiet of the highest power is the leadig coefficiet, ad the term ax is the leadig term. Notice that a polyomial is usually writte i descedig powers of the variable, ad the degree of a polyomial is the power of the leadig term. For istace 3 P x = 4x x + 5 is a polyomial of degree 3. Also, if a polyomial cosists of just a sigle term, such as Q x = 7x, the it is called a moomial. 4 Graphs of Polyomials: Polyomials of degree 0 ad 1 are liear equatios, ad their graphs are straight lies. Polyomials of degree are quadratic equatios, ad their graphs are parabolas. As the degree of the polyomial icreases beyod, the umber of possible shapes the graph ca be icreases. However, the graph of a polyomial fuctio is always a smooth cotiuous curve (o breaks, gaps, or sharp corers). Moomials of the form P x = x are the simplest polyomials.
As the figure suggest, the graph of P( x) x = has the same geeral shape as y x = whe 3 is eve, ad the same geeral shape as y = x whe is odd. However, as the degree becomes larger, the graphs become flatter aroud the origi ad steeper elsewhere. Trasformatios of Moomials: Whe graphig certai polyomial fuctios, we ca use the graphs of moomials we already kow, ad trasform them usig the techiques we leared earlier. 3 Example 1: Sketch the graph of the fuctio P( x) x of a appropriate fuctio of the form y itercepts o the graph. = + by trasformig the graph x =. Idicate all x- ad y- Based o the trasformatio techiques, we kow the graph of 3 3 P( x) = x + is the reflectio of the graph of y = x i the x-axis, shifted vertically up uits. Thus, Most polyomial fuctios caot be graphed usig trasformatios though. For istace i the polyomial fuctio 3 5 R x = x x + 1,
we caot determie easily what fuctio s graph we should perform trasformatios o to graph R ( x ). Therefore, we will eed a ew method for fidig the graphs of more complex polyomials. Ed Behavior of Polyomials: The ed behavior of a polyomial is a descriptio of what happes as x becomes large i the positive or egative directio. To describe ed behavior, we use the followig otatio: x meas x becomes large i the positive directio x meas x becomes large i the egative directio For example, the moomial y x 3 = has the ed behavior y as x ad y as x The ed behavior of a polyomial graph is determied by the term of highest degree. For 5 istace, the polyomial f x = 3x 4x + has the same ed behavior as f ( x) = 3x 5 because both are polyomials of degree 5.
Example : Determie the ed behavior of the polyomial 6 4 Q x = 6x 4x + x 3. Sice Q has eve degree ad positive leadig coefficiet, it has the followig ed behavior: y as x ad y as x Usig Zeros to Graph Polyomials: P 0 Defiitio: If is a polyomial ad c is a umber such that P c =, the we say that c is a zero of P. The followig are equivalet ways of sayig the same thig. 1. c is a zero of P. x = c is a root of the equatio P( x ) = 0 3. x c is a factor of P( x ) Whe graphig a polyomial, we wat to fid the roots of the polyomial equatio P x =. To do this, we factor the polyomial ad the use the Zero-Product Property 0 (Sectio 3.3). Remember that if P( c ) = 0, the the graph of y P( x) at x = c, so the x-itercepts of the graph are the zeros of the fuctio. Example 3: Fid the zeros of the polyomial Step 1: First we must factor R to get Step : Sice 4 = ( 4)( 3) R x x x x is a factor of x 3 is a factor of R( x) x x R x = x 7x+ 1. = has a x-itercept R x = x 7x+ 1, 4 is a zero of R, ad sice = 7 + 1, 3 is a zero of R. The followig theorem ad its cosequeces will be used to help us graph polyomials. Itermediate Value Theorem for Polyomials: If P is a polyomial fuctio ad P( a ) ad P( b ) have opposite sigs, P( c ) = 0 the there exists at least oe value c betwee a ad b for which. The figure below graphically demostrates this theorem.
Oe importat cosequece of this theorem is that betwee ay two successive zeros, the values of a polyomial are either all positive or all egative. That is, betwee two successive zeros the graph of a polyomial lies etirely above or etirely below the x- axis. So, to sketch the graph of P, we first fid all the zeros of P. The we choose test poits P x is betwee (ad to the right ad left of) successive zeros to determie whether positive or egative o each iterval determied by the zeros. Guidelies for Graphig Polyomial Fuctios: 1. Zeros: Factor the polyomial to fid all its real zeros; these are the x-itercepts of the graph.. Test Poits: Make a table of values for the polyomial. Iclude test poits to determie whether the graph of the polyomial lies above or below the x-axis o the itervals determied by the zeros. Iclude the y-itercept i the table 3. Ed Behavior: Determie the ed behavior of the polyomial. 4. Graph: Plot the itercepts ad other poits you foud i the table. Sketch a smooth curve that passes through these poits ad exhibits the required ed behavior.
Example 4: Sketch the graph of the fuctio P x = x+ 1 x graph shows all itercepts ad exhibits the proper ed behavior. Step 1: First we must fid all the real zeros of P( x ). Sice = ( + 1)( ) P x x x. Make sure your is already factored, it is easy to see the zeros are x = 1 ad x =. Therefore, the x-itercepts are x = 1 ad x =. Step : Now we will make a table of values of P( x ), makig sure we choose test poits betwee (ad to the left ad right of ) successive zeros, ad iclude the y-itercept, P 0 = 4. Step 3: Next we determie the ed behavior. If we expad P, we get P x = x x +4. Sice P is of odd degree (degree 3) ad its 3 leadig coefficiet is positive, it has the followig ed behavior: y as x ad y as x
Example 4 (Cotiued): Step 4: Fially, we plot the poits from the table ad coect the poits by a smooth curve to complete the graph. Example 5: Sketch the graph of the fuctio 4 Q x = x 4x. Make sure your graph shows all itercepts ad exhibits the proper ed behavior. Step 1: I order to fid all the real zeros of Q( x ), we must first factor it completely. 4 = 4 Q x x x x ( x 4) = = x ( x+ )( x ) Factor x Differece of Squares Step : Sice Q( x) x ( x )( x ) = + the zeros are x = 0, x = ad x =. Thus, the x-itercepts are x = 0, x = ad x =.
Example 5 (Cotiued): Step 3: Now we will make a table of values of Q( x ), makig sure we choose test poits betwee (ad to the left ad right of) successive zeros, ad iclude the y-itercept, Q 0 = 0. Step 4: Next we determie the ed behavior. Sice Q is of eve degree (degree 4) ad its leadig coefficiet is positive, it has the followig ed behavior: y as x ad y as x Step 5: Fially, we plot the poits from the table ad coect the poits by a smooth curve to complete the graph.
By examiig Example 4 ad Example 5, otice that whe c is a zero of P, ad the correspodig factor ( x c) occurs exactly m times i the factorizatio of P, the graph crosses the x-axis at c if m is odd ad does ot cross the x-axis if m is eve. Example 6: Sketch the graph of the fuctio R( x) ( x ) ( x 1) ( x ) = + +. Make sure your graph shows all itercepts ad exhibits the proper ed behavior. Step 1: The real zeros of R( x) = ( x ) ( x+ 1) ( x+ ) x = 1 are x =, ad x =. These are the x-itercepts of the graph.
Example 6 (Cotiued): Step : Now we will make a table of values of R ( x ), makig sure we choose test poits betwee (ad to the left ad right of) R 0 = 8. successive zeros, ad the y-itercept, Step 3: Sice R is of odd degree (degree 5) ad its leadig coefficiet is egative, it has the followig ed behavior: y as x ad y as x Step 4: Fially, we plot the poits from the table ad coect the poits by a smooth curve to complete the graph.