{"id":2549,"date":"2025-08-13T18:02:38","date_gmt":"2025-08-13T18:02:38","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/precalculus\/?post_type=chapter&p=2549"},"modified":"2025-10-22T23:15:40","modified_gmt":"2025-10-22T23:15:40","slug":"parabolas-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/precalculus\/chapter\/parabolas-learn-it-3\/","title":{"raw":"Parabolas: Learn It 3","rendered":"Parabolas: Learn It 3"},"content":{"raw":"

Writing Equations of Parabolas in Standard Form<\/h2>\r\nIn the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.\r\n\r\n
How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/strong>\r\n
    \r\n \t
  • Determine whether the axis of symmetry is the x<\/em>- or y<\/em>-axis.\r\n
      \r\n \t
    • If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\r\n \t
    • If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
    • Multiply [latex]4p[\/latex].<\/li>\r\n \t
    • Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\r\n<\/ul>\r\n<\/section>
      What is the equation for the parabola<\/strong> with focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex][reveal-answer q=\"570289\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"570289\"]\r\n\r\nThe focus has the form [latex]\\left(p,0\\right)[\/latex], so the equation will have the form [latex]{y}^{2}=4px[\/latex].\r\n\r\nMultiplying [latex]4p[\/latex], we have [latex]4p=4\\left(-\\frac{1}{2}\\right)=-2[\/latex]. Substituting for [latex]4p[\/latex], we have [latex]y^2=4px=2x[\/latex].\r\n\r\nTherefore, the equation for the parabola is [latex]{y}^{2}=-2x[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
      \r\n
      \r\n\r\nWhat is the equation for the parabola with focus [latex]\\left(0,\\frac{7}{2}\\right)[\/latex] and directrix [latex]y=-\\frac{7}{2}?[\/latex]\r\n\r\n[reveal-answer q=\"665981\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"665981\"]\r\n\r\n[latex]{x}^{2}=14y[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>
      [ohm_question hide_question_numbers=1]174063[\/ohm_question]<\/section>In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.\r\n\r\n
      How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/strong>\r\n
        \r\n \t
      • Determine whether the axis of symmetry is the x<\/em>- or y<\/em>-axis.\r\n
          \r\n \t
        • If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\r\n \t
        • If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Multiply [latex]4p[\/latex].<\/li>\r\n \t
        • Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\r\n<\/ul>\r\n<\/section>
          What is the equation for the parabola<\/strong> with focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex][reveal-answer q=\"47130\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"47130\"]\r\n\r\nThe focus has the form [latex]\\left(p,0\\right)[\/latex], so the equation will have the form [latex]{y}^{2}=4px[\/latex].\r\n
          \r\n
            \r\n \t
          • Multiplying [latex]4p[\/latex], we have [latex]4p=4\\left(-\\frac{1}{2}\\right)=-2[\/latex].<\/li>\r\n \t
          • Substituting for [latex]4p[\/latex], we have [latex]{y}^{2}=4px=-2x[\/latex].<\/li>\r\n<\/ul>\r\n<\/div>\r\nTherefore, the equation for the parabola is [latex]{y}^{2}=-2x[\/latex].\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
            What is the equation for the parabola with focus [latex]\\left(0,\\frac{7}{2}\\right)[\/latex] and directrix [latex]y=-\\frac{7}{2}?[\/latex][reveal-answer q=\"807025\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"807025\"][latex]{x}^{2}=14y[\/latex]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

            Writing Equations of Parabolas in Standard Form<\/h2>\n

            In the previous examples, we used the standard form equation of a parabola to calculate the locations of its key features. We can also use the calculations in reverse to write an equation for a parabola when given its key features.<\/p>\n

            How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/strong><\/p>\n
              \n
            • Determine whether the axis of symmetry is the x<\/em>– or y<\/em>-axis.\n
                \n
              • If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\n
              • If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\n<\/ul>\n<\/li>\n
              • Multiply [latex]4p[\/latex].<\/li>\n
              • Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\n<\/ul>\n<\/section>\n
                What is the equation for the parabola<\/strong> with focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex]<\/p>\n