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3.3 graphs of factorable polynomials and rational functions
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3.3 graphs of factorable polynomials and rational functions 3.3 graphs of factorable polynomials and rational functions Presentation Transcript

  • Graphs of Factorable Polynomials
  • Graphs of Factorable PolynomialsFollowing are some of the basic shapes of graphsthat we encounter often. The dotted tangent line isfor reference. Practice drawing them a few times.
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven y = x2
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven y = x4 y = x2 (-1, 1) (1, 1)
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven y = x6 y = x4 y = x2 (-1, 1) (1, 1)
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven The graphs y = –xeven y = x6 y = x4 y = x2 (-1,-1) (1,-1) y = -x2 (-1, 1) (1, 1) y = -x4 y = -x6
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven The graphs y = –xeven y = x6 y = x4 y = x2 (-1,-1) (1,-1) y = -x2 (-1, 1) (1, 1) y = -x4 y = -x6 Plot these functions and zoom in on the region around x = –1 to x = 1. Note that the graphs in between the points (1, 1) and (–1,1) drop lower as the power increases. However the graphs switch positions as they pass to the right of (1, 1) or to the left of (–1,1). ( Why?)
  • Graphs of Factorable PolynomialsWe start with the graphs of the polynomials y = xN.The graphs y = xeven The graphs y = –xeven y = x6 y = x4 y = x2 (-1,-1) (1,-1) y = -x2 (-1, 1) (1, 1) y = -x4 y = -x6 y = xE y = –xEGraphs of evenordered rootsy = xEven.
  • Graphs of Factorable PolynomialsThe graphs y = xodd y = x3
  • Graphs of Factorable PolynomialsThe graphs y = xodd y = x5 y = x3 (1, 1)(-1, -1)
  • Graphs of Factorable PolynomialsThe graphs y = xodd y = x5 y = x7 y = x3 (1, 1)(-1, -1)
  • Graphs of Factorable PolynomialsThe graphs y = xodd The graphs y = –xodd y = x5 y = -x5 y = -x3 y = -x7 y= x7 y = x3 (1, 1) (-1, 1)(-1, -1) (1,-1)
  • Graphs of Factorable PolynomialsThe graphs y = xodd The graphs y = –xodd y = x5 y = -x5 y = -x3 y = -x7 y= x7 y = x3 (1, 1) (-1, 1)(-1, -1) (1,-1) y = xD y = –xDGraphs of oddordered rootsy = xodD
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.For large |x|, the leading term anxn dominates thelower degree terms.
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.For large |x|, the leading term anxn dominates thelower degree terms. For xs such that | x | are large,the "lower degree terms" are negligible compared toa nx n.
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.For large |x|, the leading term anxn dominates thelower degree terms. For xs such that | x | are large,the "lower degree terms" are negligible compared toa nx n.Hence, for x where |x| is "large", the graph of P(x)resembles the graph y = anxn.
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.For large |x|, the leading term anxn dominates thelower degree terms. For xs such that | x | are large,the "lower degree terms" are negligible compared toa nx n.Hence, for x where |x| is "large", the graph of P(x)resembles the graphfour anxn.This means therere y = behaviors ofpolynomial-graphs to the far left or far right(as | x | becomes large).
  • Graphs of Factorable PolynomialsFacts about the graphs of polynomials:• The graphs of polynomials are unbroken curves.• Polynomial curves are smooth (no corners).• Let P(x) = anxn + lower degree terms.For large |x|, the leading term anxn dominates thelower degree terms. For xs such that | x | are large,the "lower degree terms" are negligible compared toa nx n.Hence, for x where |x| is "large", the graph of P(x)resembles the graphfour anxn.This means therere y = behaviors ofpolynomial-graphs to the far left or far right(as | x | becomes large). These behaviors are basedon the sign the leading term anxn, and whether n iseven or odd.
  • Graphs of Factorable PolynomialsI. The "Arms" of Polynomial Graphs
  • Graphs of Factorable PolynomialsI. The "Arms" of Polynomial Graphsy = +xeven + lower degree terms:
  • Graphs of Factorable PolynomialsI. The "Arms" of Polynomial Graphsy = +xeven + lower degree terms: y = –xeven + lower degree terms:
  • Graphs of Factorable PolynomialsI. The "Arms" of Polynomial Graphsy = +xeven + lower degree terms: y = –xeven + lower degree terms: y = +xodd + lower degree terms:
  • Graphs of Factorable PolynomialsI. The "Arms" of Polynomial Graphsy = +xeven + lower degree terms: y = –xeven + lower degree terms: y = +xodd + lower degree terms: y = –xodd + lower degree terms:
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:Construction of the sign-chart of polynomial P(x):
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:Construction of the sign-chart of polynomial P(x):I. Find the roots of P(x) and their order respectively.
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:Construction of the sign-chart of polynomial P(x):I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:Construction of the sign-chart of polynomial P(x):I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.III. Sample a point for its sign, use the the orders ofthe roots to extend and fill in the signs.
  • Graphs of Factorable PolynomialsFor factorable polynomials, we use the sign-charts tosketch the central portion of the graphs.Recall that given a polynomial P(x), its sign-chart isconstructed in the following manner:Construction of the sign-chart of polynomial P(x):I. Find the roots of P(x) and their order respectively.II. Draw the real line, mark off the answers from I.III. Sample a point for its sign, use the the orders ofthe roots to extend and fill in the signs.(Reminder:Across odd-ordered root, sign changesAcross even-ordered root, sign stays the same.)
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x).
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 aretwo roots of odd order.
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). y Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 are xtwo roots of odd order.The sign chart and thegraph of y = f(x) are y=(x – 4)(x+1)shown here.
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). y Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 are xtwo roots of odd order.The sign chart and thegraph of y = f(x) are y=(x – 4)(x+1)shown here.Note the sign-chart reflects the properties of the graph.
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). y Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 are xtwo roots of odd order.The sign chart and thegraph of y = f(x) are y=(x – 4)(x+1)shown here.Note the sign-chart reflects the properties of the graph.I. The graph touches or crosses the x-axis at the roots.
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). y Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 are xtwo roots of odd order.The sign chart and thegraph of y = f(x) are y=(x – 4)(x+1)shown here.Note the sign-chart reflects the properties of the graph.I. The graph touches or crosses the x-axis at the roots.II. The graph is above the x-axis where the sign is "+".
  • Graphs of Factorable PolynomialsExample A: Make the sign-chart of f(x) = x2 – 3x – 4and graph y = f(x). y Solve x2 – 3x – 4 = 0 (x – 4)(x + 1) = 0so x = 4 and x= –1 are xtwo roots of odd order.The sign chart and thegraph of y = f(x) are y=(x – 4)(x+1)shown here.Note the sign-chart reflects the properties of the graph.I. The graph touches or crosses the x-axis at the roots.II. The graph is above the x-axis where the sign is "+".III. The graph is below the x-axis where the sign is "–".
  • Graphs of Factorable PolynomialsII. The “Mid-Portions” of Polynomial Graphs
  • Graphs of Factorable PolynomialsII. The “Mid-Portions” of Polynomial GraphsGraphs of an odd ordered root (x – r)D at x = r.
  • Graphs of Factorable PolynomialsII. The “Mid-Portions” of Polynomial GraphsGraphs of an odd ordered root (x – r)D at x = r. + + r r order = 1 order = 1 y = (x – r)1 y = –(x – r)1 + + r r order = 3, 5, 7.. order = 3, 5, 7.. y = (x – r)3 or 5.. y = –(x – r)3 or 5..
  • Graphs of Factorable PolynomialsGraphs of an even ordered root at (x – r)E at x= r. x=r r + + order = 2, 4, 6 .. order = 2, 4, 6 .. y = (x – r)2 or 4.. y = –(x – r)2 or 4..If the we know the roots of a factorable polynomial,then we may construct the central portion of thegraph (the body) in the following manner using itssign chart.I. Draw the graph about each root using theinformation about the order of each root.II. Connect all the pieces together to form the graph.
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders andthe sign-chart of apolynomial,
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial,
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown.
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.Example B: Given P(x) = -x(x + 2)2(x – 3)2, identifythe roots and their orders. Make the sign-chart.Sketch the graph about each root. Connect them tocomplete the graph.
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.Example B: Given P(x) = -x(x + 2)2(x – 3)2, identifythe roots and their orders. Make the sign-chart.Sketch the graph about each root. Connect them tocomplete the graph.The roots are x = 0 of order 1,
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.Example B: Given P(x) = -x(x + 2)2(x – 3)2, identifythe roots and their orders. Make the sign-chart.Sketch the graph about each root. Connect them tocomplete the graph.The roots are x = 0 of order 1, x = -2 of order 2,
  • Graphs of Factorable PolynomialsFor example, given tworoots with their orders and order=2 order=3the sign-chart of a +polynomial, the graphsaround each root are asshown. Connect them to getthe whole graph.Example B: Given P(x) = -x(x + 2)2(x – 3)2, identifythe roots and their orders. Make the sign-chart.Sketch the graph about each root. Connect them tocomplete the graph.The roots are x = 0 of order 1, x = -2 of order 2,and x = 3 of order 2.
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2By the sign-chart and the order of each root, we drawthe graph about each root.
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2By the sign-chart and the order of each root, we drawthe graph about each root.
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2By the sign-chart and the order of each root, we drawthe graph about each root. (Note for x = 0 of order 1,the graph approximates a line going through the point.)
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2By the sign-chart and the order of each root, we drawthe graph about each root. (Note for x = 0 of order 1,the graph approximates a line going through the point.)Connect all the pieces to get the graph of P(x).
  • Graphs of Factorable PolynomialsThe sign-chart of P(x) = -x(x + 2)2(x – 3)2 is + + x = -2 x=0 x=3 order 2 order 1 order 2By the sign-chart and the order of each root, we drawthe graph about each root. (Note for x = 0 of order 1,the graph approximates a line going through the point.)Connect all the pieces to get the graph of P(x).
  • Graphs of Factorable PolynomialsNote the graph resembles y = -x5, its leading term,when viewed at a distance.
  • Graphs of Factorable PolynomialsNote the graph resembles y = -x5, its leading term,when viewed at a distance. + + -2 0 3
  • Graphs of Rational Functions
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.A rational function is factorable if both P(x) andQ(x) are factorable.
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.A rational function is factorable if both P(x) andQ(x) are factorable.In this section we study the graphs of reducedfactorable rational functions.
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.A rational function is factorable if both P(x) andQ(x) are factorable.In this section we study the graphs of reducedfactorable rational functions.The main principle of graphing these functions is thethe same as polynomials.
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.A rational function is factorable if both P(x) andQ(x) are factorable.In this section we study the graphs of reducedfactorable rational functions.The main principle of graphing these functions is thethe same as polynomials. We analyze the behaviorsand draw pieces the graphs at important locations,then complete the graphs by connecting them.
  • Graphs of Rational FunctionsRecalled that rational functions are functions of theform R(x) = P(x) where P(x) and Q(x) are Q(x)polynomials.A rational function is factorable if both P(x) andQ(x) are factorable.In this section we study the graphs of reducedfactorable rational functions.The main principle of graphing these functions is thethe same as polynomials. We analyze the behaviorsand draw pieces the graphs at important locations,then complete the graphs by connecting them.However the behaviors of rational functions are morecomplicated due to the presence of the denominators.
  • Graphs of Rational FunctionsGraph of y = ±1/xN for N = 1, 2, 3..The graph of y = 1/x has anasymptotes at x = 0 asshown here. – – – +++We also call vertical asymptotes“poles”. Therefore y = 1/x has a x=0pole of order 1 at x = 0. Graph of y = 1/xThe graph of y = 1/x2 has a poleof order 2 at x = 0 and its graphis shown here. Similar to theorders of roots, the even or oddorders of the poles determine +++ +++the behaviors of the graphs. x=0They are shown below. 2 Graph of y = 1/x
  • Graphs of Rational Functions Graphs ofodd ordered poles + +e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,..
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + +e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + +e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,.. We are examining the “Mid–Portion” of the graphs of rational functions. We will look at the “Arms” of these graphs later.
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph. VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Graphs of Graphs ofodd ordered poles even ordered poles + + + + e.g. y = 1/x1 or 3,.. e.g. y = –1/x1 or 3,.. e.g. y = 1/x2 or 4,.. e.g. y = –1/x2 or 4,..Example A: Given thefollowing information ofroots, sign-chart and rootvertical asymptotes, + +draw the graph.Graph from right to left: VA VA
  • Graphs of Rational Functions Horizontal Asymptotes
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis),
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis), the graph of a rational functionresembles the quotient of the leading terms of thenumerator and the denominator.
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis), the graph of a rational functionresembles the quotient of the leading terms of thenumerator and the denominator.Specifically, if AxN + lower degree termsR(x) = BxK + lower degree terms
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis), the graph of a rational functionresembles the quotient of the leading terms of thenumerator and the denominator.Specifically, if AxN + lower degree termsR(x) = BxK + lower degree termsthen for xs where | x | is large, the graph of R(x)resembles AxN (quotient of the leading terms). BxK
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis), the graph of a rational functionresembles the quotient of the leading terms of thenumerator and the denominator.Specifically, if AxN + lower degree termsR(x) = BxK + lower degree termsthen for xs where | x | is large, the graph of R(x)resembles AxN (quotient of the leading terms). BxKThe graph may or may not level off horizontally.
  • Graphs of Rational Functions Horizontal AsymptotesFor xs where | x | is large (i.e.. x is to the far right orfar left of the x-axis), the graph of a rational functionresembles the quotient of the leading terms of thenumerator and the denominator.Specifically, if AxN + lower degree termsR(x) = BxK + lower degree termsthen for xs where | x | is large, the graph of R(x)resembles AxN (quotient of the leading terms). BxKThe graph may or may not level off horizontally.If it does, then we have a horizontal asymptote (HA).
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior):
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K,
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B.
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. lim
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K,
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K, then the graph of R(x) has y = A/B as ahorizontal asymptote (HA).
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K, then the graph of R(x) has y = A/B as ahorizontal asymptote (HA). It is noted as x ∞ y = A/B. lim
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K, then the graph of R(x) has y = A/B as ahorizontal asymptote (HA). It is noted as x ∞ y = A/B. limIII. If N < K,
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K, then the graph of R(x) has y = A/B as ahorizontal asymptote (HA). It is noted as x ∞ y = A/B. limIII. If N < K, then the graph of R(x) has y = 0 as ahorizontal asymptote (HA) because N – K is negative.
  • Graphs of Rational FunctionsWe list all the possibilities of horizontal behavior below:Theorem (Horizontal Behavior): AxN + lower degree termsGiven that R(x) = K + lower degree terms , Bxthe graph of R(x) as x goes to the far right (x  ∞) andfar left (x -∞) behaves similar to AxN/BxK = AxN-K/B.I. If N > K, then the graph of R(x) resemble thepolynomial AxN-K/B. We write this as x ∞ y = ∞. limII. If N = K, then the graph of R(x) has y = A/B as ahorizontal asymptote (HA). It is noted as x ∞ y = A/B. limIII. If N < K, then the graph of R(x) has y = 0 as ahorizontal asymptote (HA) because N – K is negative.It is noted as lim y = 0. x ∞
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart,
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart, the shape of thegraph around the roots, and the shape of the graphalong the poles.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart, the shape of thegraph around the roots, and the shape of the graphalong the poles.Steps I and II give the "mid-section“ of the graph.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart, the shape of thegraph around the roots, and the shape of the graphalong the poles.Steps I and II give the "mid-section“ of the graph.III. (HA) Use the last theorem to determine thebehavior of the graph to the right and left as x ∞.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart, the shape of thegraph around the roots, and the shape of the graphalong the poles.Steps I and II give the "mid-section“ of the graph.III. (HA) Use the last theorem to determine thebehavior of the graph to the right and left as x ∞.The horizontal asymptote exists only if the limit exists.
  • Graphs of Rational FunctionsSteps for graphing a rational function R(x) = P(x) Q(x)I. (Roots) Find the roots of P(x) and their ordersby solving P(x) = 0.II. (Poles or VA) Find the vertical asymptotes (VA)and their orders by solving Q(x) = 0.Steps I and II give the sign-chart, the shape of thegraph around the roots, and the shape of the graphalong the poles.Steps I and II give the "mid-section“ of the graph.III. (HA) Use the last theorem to determine thebehavior of the graph to the right and left as x ∞.The horizontal asymptote exists only if the limit exists.Step III gives the “arms” of the graph.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1. x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. + – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. ++ – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. ++ – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. ++ – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. ++ – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values. ++ – + + x=2
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values.As x ∞, R(x) resemblesx2/x2 = 1, i.e. it has y = 1 ++ – + + x=2as the HA.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values.As x ∞, R(x) resemblesx2/x2 = 1, i.e. it has y = 1 ++ – + + x=2as the HA.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values.As x ∞, R(x) resemblesx2/x2 = 1, i.e. it has y = 1 ++ – + + x=2as the HA.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values.As x ∞, R(x) resemblesx2/x2 = 1, i.e. it has y = 1 ++ – + + x=2as the HA. Note thegraph stays above theHA to the far left below tothe far right.
  • Graphs of Rational FunctionsExample B.Find the roots, VA and HA, if any, of R(x) = x2 – 4x + 4 x2 – 1Draw the sign-chart and sketch the graph.For its roots, set x2 – 4x + 4 = 0, i.e.. x = 2 (ord = 2).For VA, set Q(x) = 0, i.e.. x2 – 1 = 0  x = 1.All of them have order1, so the sign changesat each of these values.As x ∞, R(x) resemblesx2/x2 = 1, i.e. it has y = 1 ++ – + + x=2as the HA. Note the (0,–4)graph stays above theHA to the far left below tothe far right. Note the y– int (0, – 4)
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. – + – + x=3
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph.As x  ∞, the graph of R(x) – + – +resembles the graph of the x=3quotient of the leading termsx2/x = x, or y = x.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph.As x  ∞, the graph of R(x) – + – +resembles the graph of the x=3quotient of the leading termsx2/x = x, or y = x.Hence there is no HA.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph.As x  ∞, the graph of R(x) – + – +resembles the graph of the x=3quotient of the leading termsx2/x = x, or y = x.Hence there is no HA.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph.As x  ∞, the graph of R(x) – + – +resembles the graph of the x=3quotient of the leading termsx2/x = x, or y = x.Hence there is no HA.
  • Graphs of Rational FunctionsExample C.Find the roots, VA and HA, if any, of R(x) = x2 – 2x – 3 x–2Draw the sign-chart and sketch the graph.Set x2 – 2x – 3 = 0  (x – 3)(x + 1) = 0so x = -1, 3 are the roots of order 1.For VA, set x – 2 = 0, i.e.. x = 2.Do the sign-chart. Construct themiddle part of the graph. (0, 3/2)As x  ∞, the graph of R(x) – + – +resembles the graph of the x=3quotient of the leading termsx2/x = x, or y = x.Hence there is no HA. Note the y– int (0, 3/2).