{"id":2633,"date":"2024-08-08T22:53:42","date_gmt":"2024-08-08T22:53:42","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2633"},"modified":"2024-11-21T22:38:37","modified_gmt":"2024-11-21T22:38:37","slug":"parabolas-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/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
In this section, we will write the equation of a parabola<\/em> in standard form, as opposed to the equation of a\u00a0quadratic<\/em>\u00a0or second degree polynomial. <\/em>The language we use when discussing the object is specific.It is true that a quadratic function forms a parabola when graphed in the plane, but here we are using the phrase\u00a0standard form of the equation of a parabola<\/em>\u00a0to indicate that we wish to describe the geometric object. When talking about this object in this context, we would naturally use the equations described below.\r\n\r\n<\/section>
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 [latex]x[\/latex]- or [latex]y[\/latex]-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 [latex]x[\/latex]-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 [latex]y[\/latex]-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=\"259208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"259208\"]The 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].\u00a0Substituting 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>
      [ohm2_question hide_question_numbers=1]24911[\/ohm2_question]<\/section>
      [ohm2_question hide_question_numbers=1]24912[\/ohm2_question]<\/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

      In this section, we will write the equation of a parabola<\/em> in standard form, as opposed to the equation of a\u00a0quadratic<\/em>\u00a0or second degree polynomial. <\/em>The language we use when discussing the object is specific.It is true that a quadratic function forms a parabola when graphed in the plane, but here we are using the phrase\u00a0standard form of the equation of a parabola<\/em>\u00a0to indicate that we wish to describe the geometric object. When talking about this object in this context, we would naturally use the equations described below.<\/p>\n<\/section>\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 [latex]x[\/latex]– or [latex]y[\/latex]-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 [latex]x[\/latex]-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 [latex]y[\/latex]-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