Graphs of Polnomials 1 Graphs of Polnomial Functions Recall that the degree of a polnomial is the highest power of the independent variable appearing in it. A polnomial can have no more roots than its degree, but it ma have less. If the degree of a polnomial is odd it must have at least one root. The degree of the first derivative of a polnomial is one less than that of the polnomial, and that of the second derivative is two less. Our skills in finding roots of polnomials and factoring them now become ver powerful curve sketching tools. Quadratic Functions These are degree polnomials of the form f() = a + b + c (where a ) which have eactl one critical point at = b a, and have no inflection points, since f () = a. Their graphs are parabolas smmetric about their ais which is the vertical line = b a.
Graphs of Polnomials Eample 1: Sketch the graph of f() =. Step 1: f () = and f () = <. The onl interesting value of f is, which divides the domain (, ) into two intervals: (, ) and (, ). Step : Put these values of into increasing order. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) f () f () + f() Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or half-frowns. 5 1 - - -1 1-1 - - - -5
Graphs of Polnomials Eample : Sketch the graph of f() = ( ). Step 1: f () = ( )( 1) = ( ) and f () = >. The onl interesting value of f is, which divides the domain (, ) into two intervals: (, ) and (, ). Step : Put these values of into increasing order. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) f () f () + f() Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or half-frowns. 9 8 7 6 5 1 1 5 6 7-1
Graphs of Polnomials Eample : Sketch the graph of f() =. Step 1: f () = = ( ( 1)) and f () = <. The onl interesting value of f is 1, which divides the domain (, ) into two intervals: (, 1) and ( 1, ). Step : Put these values of into increasing order. 1 Step : Put together as good a table as ou can. (, 1) 1 ( 1, ) f () f () + f() 5 Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 5 1 - - - -1 1-1 - - -
Graphs of Polnomials 5 Eample : Sketch the graph of f() = 9. Cubic Functions Step 1: of f is. f () = 9 and f () = 6. The interesting values of f are and, and the interesting value Step : Put these values of into increasing order.,,. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) (, ) (, ) f () + + + f () + + f() 6 6 Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 1 8 6 - - - -1 1 - - -6-8 -1
6 Graphs of Polnomials Eample 5: Sketch the graph of f() = + 9. Step 1: f () = + 9 > and f () = 6. The onl interesting value is. Step : Put these values of into increasing order. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) f () + f () + + + f() Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 5 1 - -1-1 1 - - - -5
Graphs of Polnomials 7 Eample 6: Sketch the graph of f() = +. Step 1: f () = 6 8 = ( ) and f () = 1 8 = ( ). The interesting values of f are and, and the interesting value of f is. Step : Put these values of into increasing order.,,. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which ( ) ( ) ( ) (, ),,, 5 f () + + + f () + + f() Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 1 7 7 1-1 1-1 - - -
8 Graphs of Polnomials Eample 7: Sketch the graph of f() = + 1. Step 1: f () = 9 6 = ( ) and f () = 18 6 = 6( 1). The interesting values of f are and, and the interesting value of f is 1. Step : Put these values of into increasing order., 1,. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which ( ) ( ) ( ) (, ), 1 1 1,, 5 f () + + + f () + + f() 1 Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 7 9 5 9 1 - -1 1-1 - - - -5
Graphs of Polnomials 9 Quartic Polnomials Polnomials of degree have eactl at most two inflection points(possibl none), and up to three critical points. It is not alwas possible to calculate all of these points using basic algebra. Eample 8: Sketch the graph of f() = 1 1 ( 18 + 5). Step 1: f () = 1 1 ( 6) = 1 ( 9) = 1 ( )( + ) and f () = 1 1 (1 6) = 1 1 ( ). The interesting values of f are, and, and the interesting values of f are and. Step : Put these values of into increasing order.,,,,. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) (, ) (, ) (, ) (, ) f () f () f() 7.6..5. 7.6
1 Graphs of Polnomials Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 6-5 - - - -1 1 5 - - -6-8
Graphs of Polnomials 11 Eample 9: Sketch the graph of = f() = 1 + 5. Solution: We have D(f ) =D(f ) =D(f ) = (, ), and f () = 1 1 = 1( ) = 1( )( + 1) = if = 1,, or, so Z(f ) ={ 1,, } ; f () = 6 = 1( ) = if = ( ) ± ( ) ()( ) () = 1 ± 7.5, 1. so Z(f ) = = ± 8 6 { 1 7 = ± 7 6, 1 + } 7 Thus I(f ) consists of, in increasing order, the numbers 1, 1 7 The maimal intervals contained in D(f ) I(f ) are therefore,, 1 + 7, and. (, 1), ( 1, 1 7 ), ( 1 7, ), (, 1+ 7 ), ( 1+ 7, ), and (, ).
1 Graphs of Polnomials We construct a table: (, 1) 1 ( 1, 1 ) ( ) ( 7 1 7 1 7,, 1 + ) ( ) 7 1 + 7 1 + 7, (, ) f () + + + + + + f () + + + + f() 5 7 5 1 - - -1 1-1 - -
Graphs of Polnomials 1 Quintic Polnomials Polnomials of degree 5 have at most three inflection points(at least none), and up to four critical points. Eample 1: Sketch the graph of f() = 1 5 5 + + 8 1. ) Step 1: f () = 1 + 1 + = 1 ( + + ) and f () = 8 + 6 + 8 = 8 ( + + 1. [ ( ) ] Since, b completing squares, we have f () = 1 + 1 7 +, the onl interesting value of f is. Also, b completing squares, we have f () = 8 [ ( ) ] + 55 8 +, so the onl interesting value of f is. and. 6 Step : Put these values of into increasing order.. Step : Put together as good a table as ou can showing the signs of f () and f () on the intervals into which (, ) (, ) f () + f () + + f() 1
1 Graphs of Polnomials Step : Plot the interesting points and connect them with curves which are either left or right half-smiles or halffrowns. 1 - -1 1-1 - -