GRAPHS OF POLYNOMIAL FUNCTIONS
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1 (0-) Chapter 0 Polnomial and Rational Functions. A bo of frozen specimens measures inches b inches b inches. It is wrapped in an insulating material of uniform thickness for shipment. The volume of the bo including the insulating material is 0 cubic inches (in. ). How thick is the insulation? in.. An independent marketing research agenc has determined that the best bo for breakfast cereal has a height that is inches (in.) larger than its thickness and a width that is in. larger than its thickness. If such a bo is to have a volume of in., then what should the thickness be? in. GRAPHING CALCULATOR EXERCISES Find all real roots to each polnomial equation b graphing the corresponding function and locating the -intercepts , 0.99, 0.99, ,.7,,.7, ,.,,,., GRAPHS OF POLYNOMIAL FUNCTIONS In this section Smmetr Behavior at the -Intercepts Sketching Some Graphs (, 8) f() = In Chapter we learned that the graph of a polnomial function of degree 0 or is a straight line and that the graph of a second-degree polnomial function is a parabola. In this section we will concentrate on graphs of polnomial functions of degree larger than. Smmetr Consider the graph of the quadratic function f () shown in Fig. 0.. Notice that both (, ) and (, ) are on the graph. In fact, f () f () for an value of. We get the same -coordinate whether we evaluate the function at a number or its opposite. This fact causes the graph to be smmetric about the -ais. If we folded the paper along the -ais, the two halves of the graph would coincide. Smmetric about the -Ais If f () is a function such that f () f () for an value of in its domain, then the graph of the function is said to be smmetric about the -ais. f() = FIGURE 0. (, 8) 7 8 FIGURE 0. Consider the graph of f () shown in Fig. 0.. It is not smmetric about the -ais like the graph of f (), but it has a different kind of smmetr. On the graph of f () we find the points (, 8) and (, 8). In this case f () and f () are not equal, but f () f (). Notice that the points (, 8) and (, 8) are the same distance from the origin and lie on a line through the origin. Smmetric about the Origin If f () is a function such that f () f () for an value of in its domain, then the graph of the function is said to be smmetric about the origin.
2 0. Graphs of Polnomial Functions (0-7) E X A M P L E Determining the smmetr of a graph Discuss the smmetr of the graph of each polnomial function. a) f () b) f () c) f () Solution a) Since f () () (), we have f () equal to the opposite of f (). So the graph is smmetric about the origin. b) Since f () () (), we have f () f (). So the graph is smmetric about the -ais. c) In this case f () () (). So f () f () and f () f (). This graph has neither tpe of smmetr. calculator close-up We can use graphs to check the conclusions about smmetr that were arrived at algebraicall in Eample. The graph of f () appears to be smmetric about the origin. The graph of f() appears to be smmetric about the -ais. The graph of f() does not appear to have either tpe of smmetr. 0 Behavior at the -Intercepts Each of the graphs of f () and f () has one -intercept (0, 0). If we use a positive or negative number ver close to 0 for in f (), the -coordinate is positive because the power on is even. That is wh the graph just touches the -ais at (0, 0) but does not cross the -ais there. If we use a positive number ver close to 0 for in f (), we get a positive -coordinate. If we use a negative number ver close to 0 for in f (), we get a negative -coordinate because the power on is odd. That is wh the graph crosses the -ais at (0, 0). In general, the graph crosses the -ais at an -intercept if the factor that produces that intercept has an odd power and the graph does not cross if the factor has an even power. E X A M P L E Behavior at the -intercepts Find the -intercepts and discuss the behavior of the graph of each polnomial function at its -intercepts. a) f() ( ) ( ) b) f () Solution a) The -intercepts are found b solving ( ) ( ) 0. The -intercepts are (, 0) and (, 0). The graph does not cross the -ais at (, 0) but does cross the -ais at (, 0).
3 (0-8) Chapter 0 Polnomial and Rational Functions b) The -intercepts are found b solving 0. B factoring, we get ( )( ) 0. The -intercepts are (0, 0), (, 0), and (, 0). Since each factor occurs an odd number of times, the graph crosses the -ais at each of the -intercepts. calculator close-up The graphs of the functions in Eample support the conclusions about the behavior at the -intercepts. 0 Sketching Some Graphs We can use our knowledge of smmetr and the behavior at the -intercepts to sketch some graphs of polnomial functions. To find the -intercepts, we might have to use the ideas of Section 0. to find the roots of a polnomial equation. E X A M P L E This calculator graph supports the conclusions made in Eample (a). calculator close-up Graphing a polnomial function Sketch the graph of each polnomial function. a) f () 7 b) f () Solution a) Since f () 7, the graph has neither tpe of smmetr. B Descartes rule of signs there are no negative roots to the equation 7 0, and the number of positive roots is either or. So the onl possible rational roots are and : 7 0 From the snthetic division we get f () ( )( ), and if we factor again, we get f () ( ) ( ). The -intercepts are (, 0) and (, 0). The discussion of Eample (a) applies to this function. The points (0, ), (, ), and (, 9) are also on the graph. The graph is shown in Fig. 0. on the net page. b) Since f () () (), we have f () f (). So the graph is smmetric about the -ais. We can factor the polnomial as follows: f () ( )( ) ( )( )( )( ) ( ) ( )
4 0. Graphs of Polnomial Functions (0-9) 7 This calculator graph supports the conclusions made in Eample (b). calculator close-up The -intercepts are (, 0) and (, 0). Since each factor for these intercepts has an even power, the graph does not cross the -ais at the intercepts. The points (0, ), (, 9), and (, 9) are also on the graph shown in Fig. 0.. f () = + 7 FIGURE 0. f () = + FIGURE 0. WARM-UPS True or false? Eplain our answer.. The graph of f () is smmetric about the -ais. False. The graph of is smmetric about the origin. False. If the graph of a polnomial function P() is smmetric about the -ais and P(), then P(). False. For the function f () we have f () f () for an value of. False. If f () 7, then f () 7. True. There is onl one -intercept for the graph of f (). True 7. The points (0, ) and (0, ) are the -intercepts for the graph of the function P() ( ) ( ). False 8. The -intercept for the graph of P() is (0, 9). False 9. The graph of f() ( ) ( ) does not cross the -ais at either of its -intercepts. True 0. The graph of f() 8 has three -intercepts. False 0. EXERCISES Reading and Writing After reading this section, write out the answers to these questions. Use complete sentences.. What does smmetric about the -ais mean? The graph of f() is smmetric about the -ais if f() f() for all in the domain of the function.. What does smmetric about the origin mean? The graph of f() is smmetric about the origin if f() f() for all in the domain of the function.. What is an -intercept? An -intercept is a point at which a graph intersects the -ais.. How can ou determine whether the graph of a polnomial function crosses the -ais at an -intercept? The graph of p() crosses the -ais at an -intercept if the factor corresponding to that intercept appears with an odd eponent in the prime factorization of the polnomial p() and does not cross if the eponent is even.
5 8 (0-0) Chapter 0 Polnomial and Rational Functions Discuss the smmetr of the graph of each polnomial function. See Eample.. f() Smmetric about -ais. f() Neither smmetr 7. f() Neither smmetr 8. f() 7 Smmetric about origin 9. f() Neither smmetr 0. f() 8 Smmetric about -ais. f() ( ) Neither smmetr. f() ( ) Neither smmetr. f() ( ) Smmetric about -ais. f() ( ) Smmetric about -ais. f() Smmetric about origin. f() Smmetric about origin Find the -intercepts and discuss the behavior of the graph of each polnomial function at its -intercepts. See Eample. 7. f() ( ) Does not cross at (, 0) 8. f() Crosses at, 0 9. f() Crosses at (, 0) and (, 0) 0. f() 9 Does not cross at, 0. f() Does not cross at (0, 0). f() Crosses at (, 0) and (, 0). f() ( ) ( ) Crosses at (, 0), does not cross at (, 0). f() ( ) Crosses at, 0. f() Crosses at (, 0), does not cross at (0, 0). f() Crosses at (, 0), does not cross at (, 0) 7. f() Crosses at, 0, does not cross at (, 0) 8. f() Crosses at (, 0), does not cross at (, 0) Match each polnomial function with its graph a h. 9. f() d 0. f() f. f() a. f() g. f() c. f() h. f() e. f() b (a) (b) (c) (d) (e) (f) (g) (h)
6 0. Graphs of Polnomial Functions (0-) 9 Sketch the graph of each polnomial function. See Eample. 7. f() 8. f(). f() 7. f() 9. f() 0. f() 8. f() 9. f(). f() GRAPHING CALCULATOR EXERCISES. f() ( ) ( ). f() ( ) ( ) Sketch the graph of each polnomial function. First graph the function on a calculator and use the calculator graph as a guide. 9. f() 0. f() 7 0. f() ( 0)
7 0 (0-) Chapter 0 Polnomial and Rational Functions. f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0). f() ( 0) ( 0) 0. GRAPHS OF RATIONAL FUNCTIONS In this section Domain Horizontal and Vertical Asmptotes Oblique Asmptotes Sketching the Graphs We first studied rational epressions in Chapter. In this section we will stud functions that are defined b rational epressions. Domain A rational epression was defined in Chapter as a ratio of two polnomials. If a ratio of two polnomials is used to define a function, then the function is called a rational function. Rational Function P( ) If P() and Q() are polnomials with no common factor and f () Q for ( ) Q() 0, then f () is called a rational function. The domain of a rational function is the set of all real numbers ecept those that cause the denominator to have a value of 0. E X A M P L E Domain of a rational function Find the domain of each rational function. a) f () b) g()
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