{"id":2625,"date":"2024-08-08T21:51:05","date_gmt":"2024-08-08T21:51:05","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/?post_type=chapter&p=2625"},"modified":"2025-08-15T16:58:31","modified_gmt":"2025-08-15T16:58:31","slug":"parabolas-learn-it-2","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/collegealgebra\/chapter\/parabolas-learn-it-2\/","title":{"raw":"Parabolas: Learn It 2","rendered":"Parabolas: Learn It 2"},"content":{"raw":"

Graphing Parabolas with Vertices at the Origin<\/h2>\r\n
Just as we have seen with ellipse and hyperbolas, parabolas have a standard forms. These standard forms are given below, along with their general graphs and key features.
\r\n

standard forms of parabolas with vertex [latex](0, 0)[\/latex]<\/h3>\r\nThe table below summarizes the standard features of parabolas with a vertex at the origin.\r\n\r\n\r\n\r\n\r\n\r\n
Axis of Symmetry<\/strong><\/td>\r\nEquation<\/strong><\/td>\r\nFocus<\/strong><\/td>\r\nDirectrix<\/strong><\/td>\r\nEndpoints of\u00a0Focal Diameter<\/strong><\/td>\r\n<\/tr>\r\n
[latex]x[\/latex]-axis<\/td>\r\n[latex]{y}^{2}=4px[\/latex]<\/td>\r\n[latex]\\left(p,\\text{ }0\\right)[\/latex]<\/td>\r\n[latex]x=-p[\/latex]<\/td>\r\n[latex]\\left(p,\\text{ }\\pm 2p\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
[latex]y[\/latex]-axis<\/td>\r\n[latex]{x}^{2}=4py[\/latex]<\/td>\r\n[latex]\\left(0,\\text{ }p\\right)[\/latex]<\/td>\r\n[latex]y=-p[\/latex]<\/td>\r\n[latex]\\left(\\pm 2p,\\text{ }p\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"\" (a) When [latex]p>0[\/latex] and the axis of symmetry is the x-axis, the parabola opens right. (b) When [latex]p<0[\/latex] and the axis of symmetry is the x-axis, the parabola opens left. (c) When [latex]p<0[\/latex] and the axis of symmetry is the y-axis, the parabola opens up. (d) When [latex]\\text{ }p<0\\text{ }[\/latex] and the axis of symmetry is the y-axis, the parabola opens down.[\/caption]<\/section>
Our work so far has only dealt with parabolas that open up or down, or vertical parabolas. Do you know that there is such thing as horizontal parabolas? These parabolas open either to the left or to the right. If we interchange the [latex]x[\/latex] and [latex]y[\/latex] in the vertical parabolas, we get the equations for the parabolas that open to the left or to the right.<\/section>The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and focal diameter. When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.\r\n\r\nA line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the focal diameter, these lines intersect on the axis of symmetry.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Parabola with key features labeled[\/caption]\r\n\r\n
How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.<\/strong>\r\n
    \r\n \t
  • Determine which of the standard forms applies to the given equation: [latex]{y}^{2}=4px[\/latex] or [latex]{x}^{2}=4py[\/latex].<\/li>\r\n \t
  • Use the standard form identified in Step 1 to determine the axis of symmetry, focus, equation of the directrix, and endpoints of the focal diameter.\r\n
      \r\n \t
    • If the equation is in the form [latex]{y}^{2}=4px[\/latex], then\r\n
        \r\n \t
      • the axis of symmetry is the [latex]x[\/latex]-axis, [latex]y=0[\/latex]<\/li>\r\n \t
      • set [latex]4p[\/latex] equal to the coefficient of [latex]x[\/latex] in the given equation to solve for [latex]p[\/latex]. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\r\n \t
      • use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(p,0\\right)[\/latex]<\/li>\r\n \t
      • use [latex]p[\/latex] to find the equation of the directrix, [latex]x=-p[\/latex]<\/li>\r\n \t
      • use [latex]p[\/latex] to find the endpoints of the focal diameter, [latex]\\left(p,\\pm 2p\\right)[\/latex]. Alternately, substitute [latex]x=p[\/latex] into the original equation.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
      • If the equation is in the form [latex]{x}^{2}=4py[\/latex], then\r\n
          \r\n \t
        • the axis of symmetry is the [latex]y[\/latex]-axis, [latex]x=0[\/latex]<\/li>\r\n \t
        • set [latex]4p[\/latex] equal to the coefficient of [latex]y[\/latex] in the given equation to solve for [latex]p[\/latex]. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\r\n \t
        • use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(0,p\\right)[\/latex]<\/li>\r\n \t
        • use [latex]p[\/latex] to find equation of the directrix, [latex]y=-p[\/latex]<\/li>\r\n \t
        • use [latex]p[\/latex] to find the endpoints of the focal diameter, [latex]\\left(\\pm 2p,p\\right)[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t
        • Plot the focus, directrix, and focal diameter, and draw a smooth curve to form the parabola.<\/li>\r\n<\/ul>\r\n<\/section>
          Graph [latex]{y}^{2}=24x[\/latex]. Identify and label the focus<\/strong>, directrix<\/strong>, and endpoints of the focal diameter<\/strong>.[reveal-answer q=\"885427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885427\"]The standard form that applies to the given equation is [latex]{y}^{2}=4px[\/latex]. Thus, the axis of symmetry is the [latex]x[\/latex]-axis.It follows that:\r\n
            \r\n \t
          • [latex]24=4p[\/latex], so [latex]p=6[\/latex]. Since [latex]p>0[\/latex], the parabola opens right<\/strong>\u00a0the coordinates of the focus<\/strong> are [latex]\\left(p,0\\right)=\\left(6,0\\right)[\/latex]<\/li>\r\n \t
          • the equation of the directrix<\/strong> is [latex]x=-p=-6[\/latex]<\/li>\r\n \t
          • the endpoints of the focal diameter<\/strong>\u00a0have the same [latex]x[\/latex]-coordinate at the focus. To find the endpoints, substitute [latex]x=6[\/latex] into the original equation: [latex]\\left(6,\\pm 12\\right)[\/latex]<\/li>\r\n<\/ul>\r\nNext we plot the focus, directrix, and focal diameter, and draw a smooth curve to form the parabola<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"400\"]\"\" Parabola with key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
            Graph [latex]{x}^{2}=-6y[\/latex]. Identify and label the focus<\/strong>, directrix<\/strong>, and endpoints of the focal diameter<\/strong>.[reveal-answer q=\"951555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951555\"]The standard form that applies to the given equation is [latex]{x}^{2}=4py[\/latex]. Thus, the axis of symmetry is the [latex]y[\/latex]-axis.It follows that:\r\n
              \r\n \t
            • [latex]-6=4p[\/latex], so [latex]p=-\\frac{3}{2}[\/latex]. Since [latex]p<0[\/latex], the parabola opens down<\/strong>.<\/li>\r\n \t
            • the coordinates of the focus<\/strong> are [latex]\\left(0,p\\right)=\\left(0,-\\frac{3}{2}\\right)[\/latex]<\/li>\r\n \t
            • the equation of the directrix<\/strong> is [latex]y=-p=\\frac{3}{2}[\/latex]<\/li>\r\n \t
            • the endpoints of the focal diameter<\/strong> can be found by substituting [latex]\\text{ }y=\\frac{3}{2}\\text{ }[\/latex] into the original equation, [latex]\\left(\\pm 3,-\\frac{3}{2}\\right)[\/latex]<\/li>\r\n<\/ul>\r\nNext we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the parabola<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"399\"]\"\" Parabola with key features labeled[\/caption]\r\n\r\n[\/hidden-answer]\r\n\r\n<\/section>
              [ohm2_question hide_question_numbers=1]24909[\/ohm2_question]<\/section>
              [ohm2_question hide_question_numbers=1]24910[\/ohm2_question]<\/section>
              [ohm2_question hide_question_numbers=1]24908[\/ohm2_question]<\/section><\/section>","rendered":"

              Graphing Parabolas with Vertices at the Origin<\/h2>\n
              Just as we have seen with ellipse and hyperbolas, parabolas have a standard forms. These standard forms are given below, along with their general graphs and key features.<\/p>\n
              \n

              standard forms of parabolas with vertex [latex](0, 0)[\/latex]<\/h3>\n

              The table below summarizes the standard features of parabolas with a vertex at the origin.<\/p>\n\n\n\n\n\n
              Axis of Symmetry<\/strong><\/td>\nEquation<\/strong><\/td>\nFocus<\/strong><\/td>\nDirectrix<\/strong><\/td>\nEndpoints of\u00a0Focal Diameter<\/strong><\/td>\n<\/tr>\n
              [latex]x[\/latex]-axis<\/td>\n[latex]{y}^{2}=4px[\/latex]<\/td>\n[latex]\\left(p,\\text{ }0\\right)[\/latex]<\/td>\n[latex]x=-p[\/latex]<\/td>\n[latex]\\left(p,\\text{ }\\pm 2p\\right)[\/latex]<\/td>\n<\/tr>\n
              [latex]y[\/latex]-axis<\/td>\n[latex]{x}^{2}=4py[\/latex]<\/td>\n[latex]\\left(0,\\text{ }p\\right)[\/latex]<\/td>\n[latex]y=-p[\/latex]<\/td>\n[latex]\\left(\\pm 2p,\\text{ }p\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
              \"\"
              (a) When [latex]p>0[\/latex] and the axis of symmetry is the x-axis, the parabola opens right. (b) When [latex]p<0[\/latex] and the axis of symmetry is the x-axis, the parabola opens left. (c) When [latex]p<0[\/latex] and the axis of symmetry is the y-axis, the parabola opens up. (d) When [latex]\\text{ }p<0\\text{ }[\/latex] and the axis of symmetry is the y-axis, the parabola opens down.<\/figcaption><\/figure>\n<\/section>\n
              Our work so far has only dealt with parabolas that open up or down, or vertical parabolas. Do you know that there is such thing as horizontal parabolas? These parabolas open either to the left or to the right. If we interchange the [latex]x[\/latex] and [latex]y[\/latex] in the vertical parabolas, we get the equations for the parabolas that open to the left or to the right.<\/section>\n

              The key features of a parabola are its vertex, axis of symmetry, focus, directrix, and focal diameter. When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.<\/p>\n

              A line is said to be tangent to a curve if it intersects the curve at exactly one point. If we sketch lines tangent to the parabola at the endpoints of the focal diameter, these lines intersect on the axis of symmetry.<\/p>\n

              \"\"
              Parabola with key features labeled<\/figcaption><\/figure>\n
              How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.<\/strong><\/p>\n
                \n
              • Determine which of the standard forms applies to the given equation: [latex]{y}^{2}=4px[\/latex] or [latex]{x}^{2}=4py[\/latex].<\/li>\n
              • Use the standard form identified in Step 1 to determine the axis of symmetry, focus, equation of the directrix, and endpoints of the focal diameter.\n
                  \n
                • If the equation is in the form [latex]{y}^{2}=4px[\/latex], then\n
                    \n
                  • the axis of symmetry is the [latex]x[\/latex]-axis, [latex]y=0[\/latex]<\/li>\n
                  • set [latex]4p[\/latex] equal to the coefficient of [latex]x[\/latex] in the given equation to solve for [latex]p[\/latex]. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\n
                  • use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(p,0\\right)[\/latex]<\/li>\n
                  • use [latex]p[\/latex] to find the equation of the directrix, [latex]x=-p[\/latex]<\/li>\n
                  • use [latex]p[\/latex] to find the endpoints of the focal diameter, [latex]\\left(p,\\pm 2p\\right)[\/latex]. Alternately, substitute [latex]x=p[\/latex] into the original equation.<\/li>\n<\/ul>\n<\/li>\n
                  • If the equation is in the form [latex]{x}^{2}=4py[\/latex], then\n
                      \n
                    • the axis of symmetry is the [latex]y[\/latex]-axis, [latex]x=0[\/latex]<\/li>\n
                    • set [latex]4p[\/latex] equal to the coefficient of [latex]y[\/latex] in the given equation to solve for [latex]p[\/latex]. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\n
                    • use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(0,p\\right)[\/latex]<\/li>\n
                    • use [latex]p[\/latex] to find equation of the directrix, [latex]y=-p[\/latex]<\/li>\n
                    • use [latex]p[\/latex] to find the endpoints of the focal diameter, [latex]\\left(\\pm 2p,p\\right)[\/latex]<\/li>\n<\/ul>\n<\/li>\n<\/ul>\n<\/li>\n
                    • Plot the focus, directrix, and focal diameter, and draw a smooth curve to form the parabola.<\/li>\n<\/ul>\n<\/section>\n
                      Graph [latex]{y}^{2}=24x[\/latex]. Identify and label the focus<\/strong>, directrix<\/strong>, and endpoints of the focal diameter<\/strong>.<\/p>\n