{"id":9462,"date":"2023-10-20T19:51:38","date_gmt":"2023-10-20T19:51:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=9462"},"modified":"2025-08-28T04:56:50","modified_gmt":"2025-08-28T04:56:50","slug":"polynomial-functions-learn-it-6","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/polynomial-functions-learn-it-6\/","title":{"raw":"Polynomial Functions: Learn It 6","rendered":"Polynomial Functions: Learn It 6"},"content":{"raw":"

Graphs of Polynomial Functions Cont.<\/h2>\r\n

Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like [latex]x[\/latex]-intercepts for higher-degree polynomials can get very messy and oftentimes be impossible to find by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.<\/p>\r\n

Polynomial functions of degree\u00a0[latex]2[\/latex] or more have graphs that do not have sharp corners. These types of graphs are called smooth curves<\/strong>. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<\/p>\r\n

\r\n[caption id=\"\" align=\"alignnone\" width=\"900\"]\"Graph Figure 1. On the left is a graph with a polynomial function and on the right is a graph with a function that is not a polynomial[\/caption]\r\n<\/center>\r\n

Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h3>\r\n

Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph.<\/p>\r\n

\r\n

For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\r\n<\/section>\r\n

There are two other important features of polynomials that influence the shape of its graph. The first is whether the degree is even or odd, and the second is whether the leading term is negative.<\/p>\r\n

Even Degree Polynomials<\/h4>\r\n

In the figure below, we show\u00a0the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]h\\left(x\\right)={x}^{6}[\/latex] which all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Graph Figure 2. A graph with the functions [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]h\\left(x\\right)={x}^{6}[\/latex][\/caption]\r\n<\/center>\r\n

Odd Degree Polynomials<\/h4>\r\n

The next figure\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5}[\/latex], and [latex]h\\left(x\\right)={x}^{7}[\/latex] which all have odd degrees.<\/p>\r\n

\r\n[caption id=\"\" align=\"aligncenter\" width=\"312\"]\"Graph Figure 3. A graph with the functions [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5}[\/latex], and [latex]h\\left(x\\right)={x}^{7}[\/latex][\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

Notice that one arm of the graph points down and the other points up. This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
x<\/td>\r\n[latex]f(x)=x^4[\/latex]<\/td>\r\n[latex]h(x)=x^5[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-1[\/latex]<\/td>\r\n[latex]1[\/latex]<\/td>\r\n[latex]-1[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-2[\/latex]<\/td>\r\n[latex]16[\/latex]<\/td>\r\n[latex]-32[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]-3[\/latex]<\/td>\r\n[latex]81[\/latex]<\/td>\r\n[latex]-243[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

 <\/p>\r\n

Now you try it.<\/p>\r\n

\r\n

Identify whether each graph represents a polynomial function that has a degree that is even or odd.<\/p>\r\n

    \r\n\t
  1. \"Graph<\/li>\r\n\t
  2. \"Graph<\/li>\r\n<\/ol>\r\n

    [reveal-answer q=\"657906\"]Show Solution[\/reveal-answer] [hidden-answer a=\"657906\"]<\/p>\r\n

      \r\n\t
    1. \r\n
        \r\n\t
      1. Both arms of this polynomial point upward, similar to a quadratic polynomial, therefore the degree must be even. \u00a0If you apply negative inputs to an even degree polynomial, you will get positive outputs back.<\/li>\r\n\t
      2. As the inputs of this polynomial become more negative the outputs also become negative. The only way this is possible is with an odd degree polynomial. Therefore, this polynomial must have an odd degree.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ol>\r\n

        [\/hidden-answer]<\/p>\r\n<\/section>\r\n

        \u00a0The Sign of the Leading Term<\/h4>\r\n

        What would happen if we change the sign of the leading term of an even degree polynomial? \u00a0For example, let us say that the leading term of a polynomial is [latex]-3x^4[\/latex]. \u00a0We will use a table of values to compare the outputs for a polynomial with leading term\u00a0[latex]-3x^4[\/latex] and\u00a0[latex]3x^4[\/latex].<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        x<\/td>\r\n[latex]-3x^4[\/latex]<\/td>\r\n[latex]3x^4[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]-2[\/latex]<\/td>\r\n[latex]-48[\/latex]<\/td>\r\n[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]-1[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n[latex]0[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]1[\/latex]<\/td>\r\n[latex]-3[\/latex]<\/td>\r\n[latex]3[\/latex]<\/td>\r\n<\/tr>\r\n
        [latex]2[\/latex]<\/td>\r\n[latex]-48[\/latex]<\/td>\r\n[latex]48[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

         <\/p>\r\n

        The grid below shows a plot with these points. The red points indicate a negative leading coefficient, and the blue points indicate a positive leading coefficient:<\/p>\r\n

        \r\n[caption id=\"attachment_12903\" align=\"aligncenter\" width=\"215\"]\"Grid Figure 4. A grid with the 8 points from the table[\/caption]\r\n<\/center>\r\n

         <\/p>\r\n

        The negative sign creates a reflection of [latex]3x^4[\/latex] across the [latex]x[\/latex]-axis. \u00a0The arms of a polynomial with a leading term of\u00a0[latex]-3x^4[\/latex] will point down, whereas the arms of a polynomial with leading term\u00a0[latex]3x^4[\/latex] will point up.<\/p>\r\n

        The table below summarizes all four cases mentioned.<\/p>\r\n\r\n\r\n\r\n\r\n\r\n\r\n
        Even Degree<\/th>\r\nOdd Degree<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
        \"11\"<\/a><\/td>\r\n\"12\"<\/a><\/td>\r\n<\/tr>\r\n
        \"13\"<\/a><\/td>\r\n\"14\"<\/a><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n
        \r\n

        Identify whether the leading term is positive or negative and whether the degree is even or odd for the following graphs of polynomial functions.<\/p>\r\n

          \r\n\t
        1. \"Graph<\/li>\r\n\t
        2. \"Graph<\/li>\r\n<\/ol>\r\n

          [reveal-answer q=\"257906\"]Show Solution[\/reveal-answer] [hidden-answer a=\"257906\"]<\/p>\r\n

            \r\n\t
          1. Both arms of this polynomial point in the same direction so it must have an even degree. \u00a0The leading term of the polynomial must be negative since the arms are pointing downward.<\/li>\r\n\t
          2. The arms of this polynomial point in different directions, so the degree must be odd. As the inputs get really big and positive, the outputs get really big and negative, so the leading coefficient must be negative.<\/li>\r\n<\/ol>\r\n\r\n[\/hidden-answer]<\/section>","rendered":"

            Graphs of Polynomial Functions Cont.<\/h2>\n

            Plotting polynomial functions using tables of values can be misleading because of some of the inherent characteristics of polynomials. Additionally, the algebra of finding points like [latex]x[\/latex]-intercepts for higher-degree polynomials can get very messy and oftentimes be impossible to find by hand. We have therefore developed some techniques for describing the general behavior of polynomial graphs.<\/p>\n

            Polynomial functions of degree\u00a0[latex]2[\/latex] or more have graphs that do not have sharp corners. These types of graphs are called smooth curves<\/strong>. Polynomial functions also display graphs that have no breaks. Curves with no breaks are called continuous. The figure below\u00a0shows\u00a0a graph that represents a polynomial function<\/strong> and a graph that represents a function that is not a polynomial.<\/p>\n

            \n
            \"Graph
            Figure 1. On the left is a graph with a polynomial function and on the right is a graph with a function that is not a polynomial<\/figcaption><\/figure>\n<\/div>\n

            Identifying\u00a0the Shape of the Graph of a Polynomial Function<\/h3>\n

            Knowing the degree of a polynomial function is useful in helping us predict what its graph will look like.\u00a0Because the power of the leading term is the highest, that term will grow significantly faster than the other terms as x<\/em>\u00a0gets very large or very small, so its behavior will dominate the graph.<\/p>\n

            \n

            For any polynomial, the\u00a0graph\u00a0of the polynomial will match the end behavior of the term of highest degree.<\/p>\n<\/section>\n

            There are two other important features of polynomials that influence the shape of its graph. The first is whether the degree is even or odd, and the second is whether the leading term is negative.<\/p>\n

            Even Degree Polynomials<\/h4>\n

            In the figure below, we show\u00a0the graphs of [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]h\\left(x\\right)={x}^{6}[\/latex] which all have even degrees. Notice that these graphs have similar shapes, very much like that of a\u00a0quadratic function. However, as the power increases, the graphs flatten somewhat near the origin and become steeper away from the origin.<\/p>\n

            \n
            \"Graph
            Figure 2. A graph with the functions [latex]f\\left(x\\right)={x}^{2},g\\left(x\\right)={x}^{4}[\/latex], and [latex]h\\left(x\\right)={x}^{6}[\/latex]<\/figcaption><\/figure>\n<\/div>\n

            Odd Degree Polynomials<\/h4>\n

            The next figure\u00a0shows the graphs of [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5}[\/latex], and [latex]h\\left(x\\right)={x}^{7}[\/latex] which all have odd degrees.<\/p>\n

            \n
            \"Graph
            Figure 3. A graph with the functions [latex]f\\left(x\\right)={x}^{3},g\\left(x\\right)={x}^{5}[\/latex], and [latex]h\\left(x\\right)={x}^{7}[\/latex]<\/figcaption><\/figure>\n<\/div>\n

             <\/p>\n

            Notice that one arm of the graph points down and the other points up. This is because\u00a0when your input is negative, you will get a negative output if the degree is odd.\u00a0The following table of values shows this.<\/p>\n\n\n\n\n\n\n
            x<\/td>\n[latex]f(x)=x^4[\/latex]<\/td>\n[latex]h(x)=x^5[\/latex]<\/td>\n<\/tr>\n
            [latex]-1[\/latex]<\/td>\n[latex]1[\/latex]<\/td>\n[latex]-1[\/latex]<\/td>\n<\/tr>\n
            [latex]-2[\/latex]<\/td>\n[latex]16[\/latex]<\/td>\n[latex]-32[\/latex]<\/td>\n<\/tr>\n
            [latex]-3[\/latex]<\/td>\n[latex]81[\/latex]<\/td>\n[latex]-243[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

             <\/p>\n

            Now you try it.<\/p>\n

            \n

            Identify whether each graph represents a polynomial function that has a degree that is even or odd.<\/p>\n

              \n
            1. \"Graph<\/li>\n
            2. \"Graph<\/li>\n<\/ol>\n