{"id":1311,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1311"},"modified":"2017-03-31T17:50:12","modified_gmt":"2017-03-31T17:50:12","slug":"understand-the-relationship-between-degree-and-turning-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/understand-the-relationship-between-degree-and-turning-points\/","title":{"raw":"Understand the relationship between degree and turning points","rendered":"Understand the relationship between degree and turning points"},"content":{"raw":"

In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. The graph has three turning points.\r\n<\/span><\/p>\r\n\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 11<\/b>[\/caption]\r\n

This function f<\/em>\u00a0is a 4th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\r\n\r\n

\r\n

A General Note: Interpreting Turning Points<\/h3>\r\n

A turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\r\n

A polynomial of degree n<\/em>\u00a0will have at most n<\/em> \u2013 1\u00a0turning points.<\/p>\r\n\r\n<\/div>\r\n

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Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\r\n

Find the maximum number of turning points of each polynomial function.<\/p>\r\n\r\n

  1. [latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\r\n\t
  2. [latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol><\/div>\r\n
    \r\n

    Solution<\/h3>\r\n
    1. [latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\r\n

      First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\r\n

      Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\r\n

      The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\r\n<\/li>\r\n\t

    2. [latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\r\n<\/ol>\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 12<\/b>[\/caption]\r\n

      First, identify the leading term of the polynomial function if the function were expanded.\r\n<\/span><\/p>\r\n

      Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\r\n

      The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>","rendered":"

      In addition to the end behavior, recall that we can analyze a polynomial function\u2019s local behavior. It may have a turning point where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising). Look at the graph of the polynomial function [latex]f\\left(x\\right)={x}^{4}-{x}^{3}-4{x}^{2}+4x[\/latex] in Figure 11. The graph has three turning points.
      \n<\/span><\/p>\n

      \"Graph<\/p>\n

      Figure 11<\/b><\/p>\n<\/div>\n

      This function f<\/em>\u00a0is a 4th<\/sup> degree polynomial function and has 3 turning points. The maximum number of turning points of a polynomial function is always one less than the degree of the function.<\/p>\n

      \n

      A General Note: Interpreting Turning Points<\/h3>\n

      A turning point<\/strong> is a point of the graph where the graph changes from increasing to decreasing (rising to falling) or decreasing to increasing (falling to rising).<\/p>\n

      A polynomial of degree n<\/em>\u00a0will have at most n<\/em> \u2013 1\u00a0turning points.<\/p>\n<\/div>\n

      \n
      \n
      \n

      Example 7: Finding the Maximum Number of Turning Points Using the Degree of a Polynomial Function<\/h3>\n

      Find the maximum number of turning points of each polynomial function.<\/p>\n

        \n
      1. [latex]f\\left(x\\right)=-{x}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]<\/li>\n
      2. [latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n<\/div>\n
        \n

        Solution<\/h3>\n
          \n
        1. [latex]f\\left(x\\right)=-x{}^{3}+4{x}^{5}-3{x}^{2}++1[\/latex]\n

          First, rewrite the polynomial function in descending order: [latex]f\\left(x\\right)=4{x}^{5}-{x}^{3}-3{x}^{2}++1[\/latex]<\/p>\n

          Identify the degree of the polynomial function. This polynomial function is of degree 5.<\/p>\n

          The maximum number of turning points is 5 \u2013 1 = 4.<\/p>\n<\/li>\n

        2. [latex]f\\left(x\\right)=-{\\left(x - 1\\right)}^{2}\\left(1+2{x}^{2}\\right)[\/latex]<\/li>\n<\/ol>\n
          \"Graph<\/p>\n

          Figure 12<\/b><\/p>\n<\/div>\n

          First, identify the leading term of the polynomial function if the function were expanded.
          \n<\/span><\/p>\n

          Then, identify the degree of the polynomial function. This polynomial function is of degree 4.<\/p>\n

          The maximum number of turning points is 4 \u2013 1 = 3.<\/p>\n<\/div>\n<\/div>\n<\/div>\n\n\t\t\t

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