{"id":923,"date":"2025-07-16T18:20:52","date_gmt":"2025-07-16T18:20:52","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=923"},"modified":"2026-01-13T16:11:41","modified_gmt":"2026-01-13T16:11:41","slug":"graphs-of-polynomial-functions-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/graphs-of-polynomial-functions-learn-it-4\/","title":{"raw":"Graphs of Polynomial Functions: Learn It 4","rendered":"Graphs of Polynomial Functions: Learn It 4"},"content":{"raw":"

Graphing Polynomials<\/h2>\r\n
\r\n
\r\n

Now that you understand end behavior, intercepts, and multiplicity, it's time to put all these pieces together to sketch polynomial graphs by hand.<\/p>\r\n\r\n

\r\n

How to: <\/strong>Graph Polynomials by Hand<\/p>\r\n\r\n

    \r\n \t
  1. Determine end behavior using the degree and leading coefficient<\/li>\r\n \t
  2. Find all intercepts (x-intercepts and y-intercept)<\/li>\r\n \t
  3. Identify multiplicity at each x-intercept<\/li>\r\n \t
  4. Plot intercepts and sketch local behavior at each<\/li>\r\n \t
  5. Connect with smooth curves respecting end behavior<\/li>\r\n \t
  6. Verify with additional points if needed<\/li>\r\n<\/ol>\r\n<\/section>
    Graph [latex]f(x) = (x + 2)(x - 1)(x - 4)[\/latex].Explanation:Step 1: Determine end behaviorExpanding: [latex]f(x) = x^3 - 3x^2 - 6x + 8[\/latex]Degree: 3 (odd)\r\nLeading coefficient: 1 (positive)\r\n\r\nEnd behavior: As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] As [latex]x \\to +\\infty[\/latex], [latex]f(x) \\to +\\infty[\/latex]\r\n\r\nStep 2: Find intercepts\r\n\r\n[latex]x[\/latex]-intercepts: Set each factor to zero\r\n\r\n[latex]x + 2 = 0 \\Rightarrow x = -2[\/latex]\r\n[latex]x - 1 = 0 \\Rightarrow x = 1[\/latex]\r\n[latex]x - 4 = 0 \\Rightarrow x = 4[\/latex]\r\n\r\ny-intercept: [latex]f(0) = (0 + 2)(0 - 1)(0 - 4) = (2)(-1)(-4) = 8[\/latex]\r\n\r\nIntercepts: [latex](-2, 0)[\/latex], [latex](1, 0)[\/latex], [latex](4, 0)[\/latex], and [latex](0, 8)[\/latex]\r\n\r\nStep 3: Analyze multiplicity\r\n\r\nAll factors have multiplicity 1, so the graph crosses the x-axis at each intercept.\r\n\r\nStep 4: Sketch the graph\r\nPlot intercepts: [latex](-2, 0)[\/latex], [latex](1, 0)[\/latex], [latex](4, 0)[\/latex], [latex](0, 8)[\/latex]\r\n\r\n\"\"\r\nStart from left: Graph comes from [latex]-\\infty[\/latex]\r\nCross at [latex]x = -2[\/latex]: Graph goes from negative to positive\r\n\r\n\"\"\r\nCross at [latex]x = 1[\/latex]: Graph goes from positive to negative\r\n\r\n\"\"\r\nCross at [latex]x = 4[\/latex]: Graph goes from negative to positive\r\n\r\n\"\"\r\nEnd on right: Graph goes to [latex]+\\infty[\/latex]\r\n\r\n\"\"\r\n\r\n<\/section>
    [ohm_question hide_question_numbers=1]317677[\/ohm_question]<\/section>
    Even multiplicity means the graph touches the [latex]x[\/latex]-axis and \"bounces back\" in the same direction it came from.<\/section>
    Graph [latex]f(x) = -(x + 1)^2(x - 3)[\/latex].[reveal-answer q=\"278903\"]Show Answer[\/reveal-answer]\r\n[hidden-answer a=\"278903\"]Step 1: Determine end behaviorExpanding: [latex]f(x) = -(x^3 - x^2 - 6x + 9) = -x^3 + x^2 + 6x - 9[\/latex]Degree: 3 (odd)\r\nLeading coefficient: -1 (negative)\r\n\r\nEnd behavior: As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to +\\infty[\/latex] As [latex]x \\to +\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex]\r\n\r\nStep 2: Find intercepts\r\n\r\nx-intercepts:\r\n\r\n[latex](x + 1)^2 = 0 \\Rightarrow x = -1[\/latex] (multiplicity 2)\r\n[latex]x - 3 = 0 \\Rightarrow x = 3[\/latex] (multiplicity 1)\r\n\r\ny-intercept: [latex]f(0) = -(0 + 1)^2(0 - 3) = -(1)(-3) = 3[\/latex]\r\n\r\nIntercepts: [latex](-1, 0)[\/latex], [latex](3, 0)[\/latex], and [latex](0, 3)[\/latex]\r\n\r\nStep 3: Analyze multiplicity\r\nAt [latex]x = -1[\/latex]: multiplicity 2 (even) \u2192 graph touches x-axis but doesn't cross\r\nAt [latex]x = 3[\/latex]: multiplicity 1 (odd) \u2192 graph crosses x-axis\r\n\r\nStep 4: Sketch the graph\r\nPlot intercepts: [latex](-1, 0)[\/latex], [latex](3, 0)[\/latex], [latex](0, 3)[\/latex]\r\n\r\n\"\"\r\nStart from left: Graph comes from [latex]+\\infty[\/latex]\r\nTouch at [latex]x = -1[\/latex]: Graph touches x-axis and returns to positive\r\n\r\n\"\"\r\nCross at [latex]x = 3[\/latex]: Graph crosses from positive to negative\r\n\r\n\"\"\r\nEnd on right: Graph goes to [latex]-\\infty[\/latex]\r\n\r\n\"\"\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
    [ohm_question hide_question_numbers=1]317676[\/ohm_question]<\/section>
    [ohm_question hide_question_numbers=1]317678[\/ohm_question]<\/section><\/div>\r\n<\/div>","rendered":"

    Graphing Polynomials<\/h2>\n
    \n
    \n

    Now that you understand end behavior, intercepts, and multiplicity, it’s time to put all these pieces together to sketch polynomial graphs by hand.<\/p>\n

    \n

    How to: <\/strong>Graph Polynomials by Hand<\/p>\n

      \n
    1. Determine end behavior using the degree and leading coefficient<\/li>\n
    2. Find all intercepts (x-intercepts and y-intercept)<\/li>\n
    3. Identify multiplicity at each x-intercept<\/li>\n
    4. Plot intercepts and sketch local behavior at each<\/li>\n
    5. Connect with smooth curves respecting end behavior<\/li>\n
    6. Verify with additional points if needed<\/li>\n<\/ol>\n<\/section>\n
      Graph [latex]f(x) = (x + 2)(x - 1)(x - 4)[\/latex].Explanation:Step 1: Determine end behaviorExpanding: [latex]f(x) = x^3 - 3x^2 - 6x + 8[\/latex]Degree: 3 (odd)
      \nLeading coefficient: 1 (positive)<\/p>\n

      End behavior: As [latex]x \\to -\\infty[\/latex], [latex]f(x) \\to -\\infty[\/latex] As [latex]x \\to +\\infty[\/latex], [latex]f(x) \\to +\\infty[\/latex]<\/p>\n

      Step 2: Find intercepts<\/p>\n

      [latex]x[\/latex]-intercepts: Set each factor to zero<\/p>\n

      [latex]x + 2 = 0 \\Rightarrow x = -2[\/latex]
      \n[latex]x - 1 = 0 \\Rightarrow x = 1[\/latex]
      \n[latex]x - 4 = 0 \\Rightarrow x = 4[\/latex]<\/p>\n

      y-intercept: [latex]f(0) = (0 + 2)(0 - 1)(0 - 4) = (2)(-1)(-4) = 8[\/latex]<\/p>\n

      Intercepts: [latex](-2, 0)[\/latex], [latex](1, 0)[\/latex], [latex](4, 0)[\/latex], and [latex](0, 8)[\/latex]<\/p>\n

      Step 3: Analyze multiplicity<\/p>\n

      All factors have multiplicity 1, so the graph crosses the x-axis at each intercept.<\/p>\n

      Step 4: Sketch the graph
      \nPlot intercepts: [latex](-2, 0)[\/latex], [latex](1, 0)[\/latex], [latex](4, 0)[\/latex], [latex](0, 8)[\/latex]<\/p>\n

      \"\"
      \nStart from left: Graph comes from [latex]-\\infty[\/latex]
      \nCross at [latex]x = -2[\/latex]: Graph goes from negative to positive<\/p>\n

      \"\"
      \nCross at [latex]x = 1[\/latex]: Graph goes from positive to negative<\/p>\n

      \"\"
      \nCross at [latex]x = 4[\/latex]: Graph goes from negative to positive<\/p>\n

      \"\"
      \nEnd on right: Graph goes to [latex]+\\infty[\/latex]<\/p>\n

      \"\"<\/p>\n<\/section>\n