{"id":1310,"date":"2015-11-12T18:35:30","date_gmt":"2015-11-12T18:35:30","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1310"},"modified":"2017-03-31T22:40:14","modified_gmt":"2017-03-31T22:40:14","slug":"zeros-end-behavior-and-turning-points","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-collegealgebra\/chapter\/zeros-end-behavior-and-turning-points\/","title":{"raw":"Zeros, End Behavior, and Turning Points","rendered":"Zeros, End Behavior, and Turning Points"},"content":{"raw":"

Graphs behave differently at various x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\r\n

Suppose, for example, we graph the function<\/p>\r\n\r\n

[latex]f\\left(x\\right)=\\left(x+3\\right){\\left(x - 2\\right)}^{2}{\\left(x+1\\right)}^{3}[\/latex].<\/div>\r\nNotice in Figure 7\u00a0that the behavior of the function at each of the x<\/em>-intercepts is different.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 7.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.[\/caption]\r\n

The x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\r\n

The x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\r\n\r\n

[latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\r\n

The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\r\n

The x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\r\nFor zeros<\/strong> with even multiplicities, the graphs touch<\/em> or are tangent to the x<\/em>-axis. For zeros with odd multiplicities, the graphs cross<\/em> or intersect the x<\/em>-axis. See Figure 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"GraphFigure 8<\/b>[\/caption]\r\n

For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x<\/em>-axis.<\/p>\r\n

For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x<\/em>-axis.<\/p>\r\n\r\n

\r\n

A General Note: Graphical Behavior of Polynomials at x<\/em>-Intercepts<\/h3>\r\n

If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the x<\/em>-intercept h\u00a0<\/em>is determined by the power p<\/em>. We say that [latex]x=h[\/latex] is a zero of multiplicity<\/strong> p<\/em>.<\/p>\r\n

The graph of a polynomial function will touch the x<\/em>-axis at zeros with even multiplicities. The graph will cross the x<\/em>-axis at zeros with odd multiplicities.<\/p>\r\n

The sum of the multiplicities is the degree of the polynomial function.<\/p>\r\n\r\n<\/div>\r\n

\r\n

How To: Given a graph of a polynomial function of degree n<\/i>, identify the zeros and their multiplicities.<\/h3>\r\n
  1. If the graph crosses the x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\r\n\t
  2. If the graph touches the x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\r\n\t
  3. If the graph crosses the x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\r\n\t
  4. The sum of the multiplicities is n<\/em>.<\/li>\r\n<\/ol><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Example 6: Identifying Zeros and Their Multiplicities<\/h3>\r\nUse the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"ThreeFigure 9<\/b>[\/caption]\r\n\r\n<\/div>\r\n
    \r\n

    Solution<\/h3>\r\n

    The polynomial function is of degree n<\/em>. The sum of the multiplicities must be n<\/em>.<\/p>\r\n

    Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\r\n

    The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\r\n

    The last zero occurs at [latex]x=4[\/latex]. The graph crosses the x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/p>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n

    \r\n

    Try It 2<\/h3>\r\nUse the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"GraphFigure 10<\/b>[\/caption]\r\n\r\nSolution<\/a>\r\n\r\n<\/div>\r\n\u00a0\r\n

    Determine end behavior<\/h3>\r\n

    As we have already learned, the behavior of a graph of a polynomial function<\/strong> of the form<\/p>\r\n\r\n

    [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\r\n

    will either ultimately rise or fall as x<\/em>\u00a0increases without bound and will either rise or fall as x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\r\n

    Recall that we call this behavior the end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\r\n\r\n
    Even Degree<\/th>\r\nOdd Degree<\/th>\r\n<\/tr><\/thead>
    \"11\"<\/a><\/td>\r\n\"12\"<\/a><\/td>\r\n<\/tr>
    \"13\"<\/a><\/td>\r\n\"14\"<\/a><\/td>\r\n<\/tr><\/tbody><\/table>\r\n\u00a0\r\n

    Understand the relationship between degree and turning points<\/h3>\r\n

    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

    \r\n
    \r\n
    \r\n

    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":"

        Graphs behave differently at various x<\/em>-intercepts. Sometimes, the graph will cross over the horizontal axis at an intercept. Other times, the graph will touch the horizontal axis and bounce off.<\/p>\n

        Suppose, for example, we graph the function<\/p>\n

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

        Notice in Figure 7\u00a0that the behavior of the function at each of the x<\/em>-intercepts is different.<\/p>\n

        \"Graph<\/p>\n

        Figure 7.<\/b> Identifying the behavior of the graph at an x-intercept by examining the multiplicity of the zero.<\/p>\n<\/div>\n

        The x<\/em>-intercept [latex]x=-3[\/latex]\u00a0is the solution of equation [latex]\\left(x+3\\right)=0[\/latex]. The graph passes directly through the x<\/em>-intercept at [latex]x=-3[\/latex]. The factor is linear (has a degree of 1), so the behavior near the intercept is like that of a line\u2014it passes directly through the intercept. We call this a single zero because the zero corresponds to a single factor of the function.<\/p>\n

        The x<\/em>-intercept [latex]x=2[\/latex] is the repeated solution of equation [latex]{\\left(x - 2\\right)}^{2}=0[\/latex]. The graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic\u2014it bounces off of the horizontal axis at the intercept.<\/p>\n

        [latex]{\\left(x - 2\\right)}^{2}=\\left(x - 2\\right)\\left(x - 2\\right)[\/latex]<\/div>\n

        The factor is repeated, that is, the factor [latex]\\left(x - 2\\right)[\/latex] appears twice. The number of times a given factor appears in the factored form of the equation of a polynomial is called the multiplicity<\/strong>. The zero associated with this factor, [latex]x=2[\/latex], has multiplicity 2 because the factor [latex]\\left(x - 2\\right)[\/latex] occurs twice.<\/p>\n

        The x-<\/em>intercept [latex]x=-1[\/latex] is the repeated solution of factor [latex]{\\left(x+1\\right)}^{3}=0[\/latex]. The graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic\u2014with the same S-shape near the intercept as the toolkit function [latex]f\\left(x\\right)={x}^{3}[\/latex]. We call this a triple zero, or a zero with multiplicity 3.<\/p>\n

        For zeros<\/strong> with even multiplicities, the graphs touch<\/em> or are tangent to the x<\/em>-axis. For zeros with odd multiplicities, the graphs cross<\/em> or intersect the x<\/em>-axis. See Figure 8\u00a0for examples of graphs of polynomial functions with multiplicity 1, 2, and 3.<\/p>\n

        \"Graph<\/p>\n

        Figure 8<\/b><\/p>\n<\/div>\n

        For higher even powers, such as 4, 6, and 8, the graph will still touch and bounce off of the horizontal axis but, for each increasing even power, the graph will appear flatter as it approaches and leaves the x<\/em>-axis.<\/p>\n

        For higher odd powers, such as 5, 7, and 9, the graph will still cross through the horizontal axis, but for each increasing odd power, the graph will appear flatter as it approaches and leaves the x<\/em>-axis.<\/p>\n

        \n

        A General Note: Graphical Behavior of Polynomials at x<\/em>-Intercepts<\/h3>\n

        If a polynomial contains a factor of the form [latex]{\\left(x-h\\right)}^{p}[\/latex], the behavior near the x<\/em>-intercept h\u00a0<\/em>is determined by the power p<\/em>. We say that [latex]x=h[\/latex] is a zero of multiplicity<\/strong> p<\/em>.<\/p>\n

        The graph of a polynomial function will touch the x<\/em>-axis at zeros with even multiplicities. The graph will cross the x<\/em>-axis at zeros with odd multiplicities.<\/p>\n

        The sum of the multiplicities is the degree of the polynomial function.<\/p>\n<\/div>\n

        \n

        How To: Given a graph of a polynomial function of degree n<\/i>, identify the zeros and their multiplicities.<\/h3>\n
          \n
        1. If the graph crosses the x<\/em>-axis and appears almost linear at the intercept, it is a single zero.<\/li>\n
        2. If the graph touches the x<\/em>-axis and bounces off of the axis, it is a zero with even multiplicity.<\/li>\n
        3. If the graph crosses the x<\/em>-axis at a zero, it is a zero with odd multiplicity.<\/li>\n
        4. The sum of the multiplicities is n<\/em>.<\/li>\n<\/ol>\n<\/div>\n
          \n
          \n
          \n

          Example 6: Identifying Zeros and Their Multiplicities<\/h3>\n

          Use the graph of the function of degree 6 to identify the zeros of the function and their possible multiplicities.<\/p>\n

          \"Three<\/p>\n

          Figure 9<\/b><\/p>\n<\/div>\n<\/div>\n

          \n

          Solution<\/h3>\n

          The polynomial function is of degree n<\/em>. The sum of the multiplicities must be n<\/em>.<\/p>\n

          Starting from the left, the first zero occurs at [latex]x=-3[\/latex]. The graph touches the x<\/em>-axis, so the multiplicity of the zero must be even. The zero of \u20133 has multiplicity 2.<\/p>\n

          The next zero occurs at [latex]x=-1[\/latex]. The graph looks almost linear at this point. This is a single zero of multiplicity 1.<\/p>\n

          The last zero occurs at [latex]x=4[\/latex]. The graph crosses the x<\/em>-axis, so the multiplicity of the zero must be odd. We know that the multiplicity is likely 3 and that the sum of the multiplicities is likely 6.<\/p>\n<\/div>\n<\/div>\n<\/div>\n

          \n

          Try It 2<\/h3>\n

          Use the graph of the function of degree 5 to identify the zeros of the function and their multiplicities.<\/p>\n

          \"Graph<\/p>\n

          Figure 10<\/b><\/p>\n<\/div>\n

          Solution<\/a><\/p>\n<\/div>\n

          \u00a0<\/p>\n

          Determine end behavior<\/h3>\n

          As we have already learned, the behavior of a graph of a polynomial function<\/strong> of the form<\/p>\n

          [latex]f\\left(x\\right)={a}_{n}{x}^{n}+{a}_{n - 1}{x}^{n - 1}+...+{a}_{1}x+{a}_{0}[\/latex]<\/div>\n

          will either ultimately rise or fall as x<\/em>\u00a0increases without bound and will either rise or fall as x\u00a0<\/em>decreases without bound. This is because for very large inputs, say 100 or 1,000, the leading term dominates the size of the output. The same is true for very small inputs, say \u2013100 or \u20131,000.<\/p>\n

          Recall that we call this behavior the end behavior<\/em> of a function. As we pointed out when discussing quadratic equations, when the leading term of a polynomial function, [latex]{a}_{n}{x}^{n}[\/latex], is an even power function, as x<\/em>\u00a0increases or decreases without bound, [latex]f\\left(x\\right)[\/latex] increases without bound. When the leading term is an odd power function, as\u00a0x<\/em>\u00a0decreases without bound, [latex]f\\left(x\\right)[\/latex] also decreases without bound; as x<\/em>\u00a0increases without bound, [latex]f\\left(x\\right)[\/latex] also increases without bound. If the leading term is negative, it will change the direction of the end behavior. The table below\u00a0summarizes all four cases.<\/p>\n\n\n\n\n\n\n
          Even Degree<\/th>\nOdd Degree<\/th>\n<\/tr>\n<\/thead>\n
          \"11\"<\/a><\/td>\n\"12\"<\/a><\/td>\n<\/tr>\n
          \"13\"<\/a><\/td>\n\"14\"<\/a><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

          \u00a0<\/p>\n

          Understand the relationship between degree and turning points<\/h3>\n

          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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