{"id":2393,"date":"2016-11-03T21:04:45","date_gmt":"2016-11-03T21:04:45","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/?post_type=chapter&#038;p=2393"},"modified":"2025-10-15T22:40:46","modified_gmt":"2025-10-15T22:40:46","slug":"graphing-parabolas-with-vertices-at-the-origin","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebra\/chapter\/graphing-parabolas-with-vertices-at-the-origin\/","title":{"raw":"Parabolas with Vertices at the Origin","rendered":"Parabolas with Vertices at the Origin"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Identify and label the focus, directrix, and endpoints of the focal diameter of a parabola.<\/li>\r\n \t<li>Write the equation of a parabola given a focus and directrix.<\/li>\r\n<\/ul>\r\n<\/div>\r\nIn The Ellipse we saw that an <strong>ellipse<\/strong> is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a <strong>parabola<\/strong>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204534\/CNX_Precalc_Figure_10_03_0022.jpg\" alt=\"\" width=\"487\" height=\"425\" \/> Parabola[\/caption]\r\n\r\nLike the ellipse and <strong>hyperbola<\/strong>, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane that are the same distance from a fixed line, called the <strong>directrix<\/strong>, and a fixed point (the <strong>focus<\/strong>) not on the directrix.\r\n\r\nWe previously learned about a parabola\u2019s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.\r\n\r\nThe line segment that passes through the focus and is parallel to the directrix is called the <strong>latus rectum,\u00a0<\/strong>also called the\u00a0<strong>focal diameter<\/strong>. The endpoints of the focal diameter\u00a0lie on the curve. By definition, the distance [latex]d[\/latex] from the focus to any point [latex]P[\/latex] on the parabola is equal to the distance from [latex]P[\/latex] to the directrix.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204536\/CNX_Precalc_Figure_10_03_003n2.jpg\" alt=\"\" width=\"487\" height=\"291\" \/> Key features of the parabola[\/caption]\r\n\r\nTo work with parabolas in the <strong>coordinate plane<\/strong>, we consider two cases: those with a vertex at the origin and those with a <strong>vertex<\/strong> at a point other than the origin. We begin with the former.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204538\/CNX_Precalc_Figure_10_03_0182.jpg\" width=\"487\" height=\"292\" \/>\r\n\r\nLet [latex]\\left(x,y\\right)[\/latex] be a point on the parabola with vertex [latex]\\left(0,0\\right)[\/latex], focus [latex]\\left(0,p\\right)[\/latex], and directrix [latex]y= -p[\/latex]\u00a0as shown in Figure 4. The distance [latex]d[\/latex] from point [latex]\\left(x,y\\right)[\/latex] to point [latex]\\left(x,-p\\right)[\/latex]\u00a0on the directrix is the difference of the <em>y<\/em>-values: [latex]d=y+p[\/latex]. The distance from the focus [latex]\\left(0,p\\right)[\/latex] to the point [latex]\\left(x,y\\right)[\/latex] is also equal to [latex]d[\/latex] and can be expressed using the <strong>distance formula<\/strong>.\r\n<p style=\"text-align: center\">[latex]\\begin{align}d&amp;=\\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y-p\\right)}^{2}} \\\\ &amp;=\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}} \\end{align}[\/latex]<\/p>\r\nSet the two expressions for [latex]d[\/latex] equal to each other and solve for [latex]y[\/latex] to derive the equation of the parabola. We do this because the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(0,p\\right)[\/latex] equals the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(x, -p\\right)[\/latex].\r\n<p style=\"text-align: center\">[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]<\/p>\r\nWe then square both sides of the equation, expand the squared terms, and simplify by combining like terms.\r\n<p style=\"text-align: center\">[latex]\\begin{gathered}{x}^{2}+{\\left(y-p\\right)}^{2}={\\left(y+p\\right)}^{2} \\\\ {x}^{2}+{y}^{2}-2py+{p}^{2}={y}^{2}+2py+{p}^{2}\\\\ {x}^{2}-2py=2py \\\\ {x}^{2}=4py\\end{gathered}[\/latex]<\/p>\r\nThe equations of parabolas with vertex [latex]\\left(0,0\\right)[\/latex] are [latex]{y}^{2}=4px[\/latex] when the <em>x<\/em>-axis is the axis of symmetry and [latex]{x}^{2}=4py[\/latex] when the <em>y<\/em>-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.\r\n<div class=\"textbox\">\r\n<h3>A General Note: Standard Forms of Parabolas with Vertex (0, 0)<\/h3>\r\nThe table below summarizes the standard features of parabolas with a vertex at the origin.\r\n<table summary=\"..\">\r\n<tbody>\r\n<tr>\r\n<td><strong>Axis of Symmetry<\/strong><\/td>\r\n<td><strong>Equation<\/strong><\/td>\r\n<td><strong>Focus<\/strong><\/td>\r\n<td><strong>Directrix<\/strong><\/td>\r\n<td><strong>Endpoints of\u00a0Focal Diameter<\/strong><\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>x<\/em>-axis<\/td>\r\n<td>[latex]{y}^{2}=4px[\/latex]<\/td>\r\n<td>[latex]\\left(p,\\text{ }0\\right)[\/latex]<\/td>\r\n<td>[latex]x=-p[\/latex]<\/td>\r\n<td>[latex]\\left(p,\\text{ }\\pm 2p\\right)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td><em>y<\/em>-axis<\/td>\r\n<td>[latex]{x}^{2}=4py[\/latex]<\/td>\r\n<td>[latex]\\left(0,\\text{ }p\\right)[\/latex]<\/td>\r\n<td>[latex]y=-p[\/latex]<\/td>\r\n<td>[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\"]<img src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204540\/CNX_Precalc_Figure_10_03_004n2.jpg\" alt=\"\" width=\"975\" height=\"721\" \/> (a) When [latex]p&gt;0[\/latex] and the axis of symmetry is the x-axis, the parabola opens right. (b) When [latex]p&lt;0[\/latex] and the axis of symmetry is the x-axis, the parabola opens left. (c) When [latex]p&lt;0[\/latex] and the axis of symmetry is the y-axis, the parabola opens up. (d) When [latex]\\text{ }p&lt;0\\text{ }[\/latex] and the axis of symmetry is the y-axis, the parabola opens down.[\/caption]<\/div>\r\nThe 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<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204543\/CNX_Precalc_Figure_10_03_0052.jpg\" width=\"487\" height=\"514\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.<\/h3>\r\n<ul>\r\n \t<li>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<li>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<ul>\r\n \t<li>If the equation is in the form [latex]{y}^{2}=4px[\/latex], then\r\n<ul>\r\n \t<li>the axis of symmetry is the <em>x<\/em>-axis, [latex]y=0[\/latex]<\/li>\r\n \t<li>set [latex]4p[\/latex] equal to the coefficient of <em>x <\/em>in the given equation to solve for [latex]p[\/latex]. If [latex]p&gt;0[\/latex], the parabola opens right. If [latex]p&lt;0[\/latex], the parabola opens left.<\/li>\r\n \t<li>use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(p,0\\right)[\/latex]<\/li>\r\n \t<li>use [latex]p[\/latex] to find the equation of the directrix, [latex]x=-p[\/latex]<\/li>\r\n \t<li>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<li>If the equation is in the form [latex]{x}^{2}=4py[\/latex], then\r\n<ul>\r\n \t<li>the axis of symmetry is the <em>y<\/em>-axis, [latex]x=0[\/latex]<\/li>\r\n \t<li>set [latex]4p[\/latex] equal to the coefficient of <em>y <\/em>in the given equation to solve for [latex]p[\/latex]. If [latex]p&gt;0[\/latex], the parabola opens up. If [latex]p&lt;0[\/latex], the parabola opens down.<\/li>\r\n \t<li>use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(0,p\\right)[\/latex]<\/li>\r\n \t<li>use [latex]p[\/latex] to find equation of the directrix, [latex]y=-p[\/latex]<\/li>\r\n \t<li>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<li>Plot the focus, directrix, and focal diameter, and draw a smooth curve to form the parabola.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Parabola with Vertex (0, 0) and the <em>x<\/em>-axis as the Axis of Symmetry<\/h3>\r\nGraph [latex]{y}^{2}=24x[\/latex]. Identify and label the <strong>focus<\/strong>, <strong>directrix<\/strong>, and endpoints of the <strong>focal diameter<\/strong>.\r\n\r\n[reveal-answer q=\"885427\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"885427\"]\r\n\r\nThe standard form that applies to the given equation is [latex]{y}^{2}=4px[\/latex]. Thus, the axis of symmetry is the <em>x<\/em>-axis. It follows that:\r\n<ul>\r\n \t<li>[latex]24=4p[\/latex], so [latex]p=6[\/latex]. Since [latex]p&gt;0[\/latex], the parabola opens right\u00a0the coordinates of the focus are [latex]\\left(p,0\\right)=\\left(6,0\\right)[\/latex]<\/li>\r\n \t<li>the equation of the directrix is [latex]x=-p=-6[\/latex]<\/li>\r\n \t<li>the endpoints of the focal diameter\u00a0have the same <em>x<\/em>-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 <strong>parabola<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204545\/CNX_Precalc_Figure_10_03_0192.jpg\" width=\"487\" height=\"376\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph [latex]{y}^{2}=-16x[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.\r\n\r\n[reveal-answer q=\"277220\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"277220\"]\r\n\r\nFocus: [latex]\\left(-4,0\\right)[\/latex]; Directrix: [latex]x=4[\/latex]; Endpoints of the latus rectum: [latex]\\left(-4,\\pm 8\\right)[\/latex]\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\"><img class=\"aligncenter size-full wp-image-3276\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23511&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"550\"><\/iframe>\r\n\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Graphing a Parabola with Vertex (0, 0) and the <em>y<\/em>-axis as the Axis of Symmetry<\/h3>\r\nGraph [latex]{x}^{2}=-6y[\/latex]. Identify and label the <strong>focus<\/strong>, <strong>directrix<\/strong>, and endpoints of the <strong>focal diameter<\/strong>.\r\n\r\n[reveal-answer q=\"951555\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"951555\"]\r\n\r\nThe standard form that applies to the given equation is [latex]{x}^{2}=4py[\/latex]. Thus, the axis of symmetry is the <em>y<\/em>-axis. It follows that:\r\n<ul>\r\n \t<li>[latex]-6=4p[\/latex], so [latex]p=-\\frac{3}{2}[\/latex]. Since [latex]p&lt;0[\/latex], the parabola opens down.<\/li>\r\n \t<li>the coordinates of the focus are [latex]\\left(0,p\\right)=\\left(0,-\\frac{3}{2}\\right)[\/latex]<\/li>\r\n \t<li>the equation of the directrix is [latex]y=-p=\\frac{3}{2}[\/latex]<\/li>\r\n \t<li>the endpoints of the focal diameter 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 <strong>parabola<\/strong>.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204547\/CNX_Precalc_Figure_10_03_007n2.jpg\" width=\"487\" height=\"327\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nGraph [latex]{x}^{2}=8y[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.\r\n\r\n[reveal-answer q=\"824847\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"824847\"]\r\n\r\nFocus: [latex]\\left(0,2\\right)[\/latex]; Directrix: [latex]y=-2[\/latex]; Endpoints of the latus rectum: [latex]\\left(\\pm 4,2\\right)[\/latex].\r\n\r\n<a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\"><img class=\"aligncenter size-full wp-image-3277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\" alt=\"\" width=\"487\" height=\"365\" \/><\/a>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n\r\n<h2>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<div class=\"textbox\">\r\n<h3>How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/h3>\r\n<ul>\r\n \t<li>Determine whether the axis of symmetry is the <em>x<\/em>- or <em>y<\/em>-axis.\r\n<ul>\r\n \t<li>If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the <em>x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\r\n \t<li>If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the <em>y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Multiply [latex]4p[\/latex].<\/li>\r\n \t<li>Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix<\/h3>\r\nWhat is the equation for the <strong>parabola<\/strong> with <strong>focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and <strong>directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex]\r\n\r\n[reveal-answer q=\"259208\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"259208\"]\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].\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<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\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=\"886076\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"886076\"]\r\n\r\n[latex]{x}^{2}=14y[\/latex]\r\n\r\n[\/hidden-answer]\r\n<iframe id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29711&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><span style=\"width: 0px;overflow: hidden;line-height: 0\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe>\r\n\r\n<\/div>\r\n&nbsp;\r\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Identify and label the focus, directrix, and endpoints of the focal diameter of a parabola.<\/li>\n<li>Write the equation of a parabola given a focus and directrix.<\/li>\n<\/ul>\n<\/div>\n<p>In The Ellipse we saw that an <strong>ellipse<\/strong> is formed when a plane cuts through a right circular cone. If the plane is parallel to the edge of the cone, an unbounded curve is formed. This curve is a <strong>parabola<\/strong>.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204534\/CNX_Precalc_Figure_10_03_0022.jpg\" alt=\"\" width=\"487\" height=\"425\" \/><\/p>\n<p class=\"wp-caption-text\">Parabola<\/p>\n<\/div>\n<p>Like the ellipse and <strong>hyperbola<\/strong>, the parabola can also be defined by a set of points in the coordinate plane. A parabola is the set of all points [latex]\\left(x,y\\right)[\/latex] in a plane that are the same distance from a fixed line, called the <strong>directrix<\/strong>, and a fixed point (the <strong>focus<\/strong>) not on the directrix.<\/p>\n<p>We previously learned about a parabola\u2019s vertex and axis of symmetry. Now we extend the discussion to include other key features of the parabola. Notice that the axis of symmetry passes through the focus and vertex and is perpendicular to the directrix. The vertex is the midpoint between the directrix and the focus.<\/p>\n<p>The line segment that passes through the focus and is parallel to the directrix is called the <strong>latus rectum,\u00a0<\/strong>also called the\u00a0<strong>focal diameter<\/strong>. The endpoints of the focal diameter\u00a0lie on the curve. By definition, the distance [latex]d[\/latex] from the focus to any point [latex]P[\/latex] on the parabola is equal to the distance from [latex]P[\/latex] to the directrix.<\/p>\n<div style=\"width: 497px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204536\/CNX_Precalc_Figure_10_03_003n2.jpg\" alt=\"\" width=\"487\" height=\"291\" \/><\/p>\n<p class=\"wp-caption-text\">Key features of the parabola<\/p>\n<\/div>\n<p>To work with parabolas in the <strong>coordinate plane<\/strong>, we consider two cases: those with a vertex at the origin and those with a <strong>vertex<\/strong> at a point other than the origin. We begin with the former.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204538\/CNX_Precalc_Figure_10_03_0182.jpg\" width=\"487\" height=\"292\" alt=\"image\" \/><\/p>\n<p>Let [latex]\\left(x,y\\right)[\/latex] be a point on the parabola with vertex [latex]\\left(0,0\\right)[\/latex], focus [latex]\\left(0,p\\right)[\/latex], and directrix [latex]y= -p[\/latex]\u00a0as shown in Figure 4. The distance [latex]d[\/latex] from point [latex]\\left(x,y\\right)[\/latex] to point [latex]\\left(x,-p\\right)[\/latex]\u00a0on the directrix is the difference of the <em>y<\/em>-values: [latex]d=y+p[\/latex]. The distance from the focus [latex]\\left(0,p\\right)[\/latex] to the point [latex]\\left(x,y\\right)[\/latex] is also equal to [latex]d[\/latex] and can be expressed using the <strong>distance formula<\/strong>.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{align}d&=\\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y-p\\right)}^{2}} \\\\ &=\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}} \\end{align}[\/latex]<\/p>\n<p>Set the two expressions for [latex]d[\/latex] equal to each other and solve for [latex]y[\/latex] to derive the equation of the parabola. We do this because the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(0,p\\right)[\/latex] equals the distance from [latex]\\left(x,y\\right)[\/latex] to [latex]\\left(x, -p\\right)[\/latex].<\/p>\n<p style=\"text-align: center\">[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]<\/p>\n<p>We then square both sides of the equation, expand the squared terms, and simplify by combining like terms.<\/p>\n<p style=\"text-align: center\">[latex]\\begin{gathered}{x}^{2}+{\\left(y-p\\right)}^{2}={\\left(y+p\\right)}^{2} \\\\ {x}^{2}+{y}^{2}-2py+{p}^{2}={y}^{2}+2py+{p}^{2}\\\\ {x}^{2}-2py=2py \\\\ {x}^{2}=4py\\end{gathered}[\/latex]<\/p>\n<p>The equations of parabolas with vertex [latex]\\left(0,0\\right)[\/latex] are [latex]{y}^{2}=4px[\/latex] when the <em>x<\/em>-axis is the axis of symmetry and [latex]{x}^{2}=4py[\/latex] when the <em>y<\/em>-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.<\/p>\n<div class=\"textbox\">\n<h3>A General Note: Standard Forms of Parabolas with Vertex (0, 0)<\/h3>\n<p>The table below summarizes the standard features of parabolas with a vertex at the origin.<\/p>\n<table summary=\"..\">\n<tbody>\n<tr>\n<td><strong>Axis of Symmetry<\/strong><\/td>\n<td><strong>Equation<\/strong><\/td>\n<td><strong>Focus<\/strong><\/td>\n<td><strong>Directrix<\/strong><\/td>\n<td><strong>Endpoints of\u00a0Focal Diameter<\/strong><\/td>\n<\/tr>\n<tr>\n<td><em>x<\/em>-axis<\/td>\n<td>[latex]{y}^{2}=4px[\/latex]<\/td>\n<td>[latex]\\left(p,\\text{ }0\\right)[\/latex]<\/td>\n<td>[latex]x=-p[\/latex]<\/td>\n<td>[latex]\\left(p,\\text{ }\\pm 2p\\right)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td><em>y<\/em>-axis<\/td>\n<td>[latex]{x}^{2}=4py[\/latex]<\/td>\n<td>[latex]\\left(0,\\text{ }p\\right)[\/latex]<\/td>\n<td>[latex]y=-p[\/latex]<\/td>\n<td>[latex]\\left(\\pm 2p,\\text{ }p\\right)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<div style=\"width: 985px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204540\/CNX_Precalc_Figure_10_03_004n2.jpg\" alt=\"\" width=\"975\" height=\"721\" \/><\/p>\n<p class=\"wp-caption-text\">(a) When [latex]p&gt;0[\/latex] and the axis of symmetry is the x-axis, the parabola opens right. (b) When [latex]p&lt;0[\/latex] and the axis of symmetry is the x-axis, the parabola opens left. (c) When [latex]p&lt;0[\/latex] and the axis of symmetry is the y-axis, the parabola opens up. (d) When [latex]\\text{ }p&lt;0\\text{ }[\/latex] and the axis of symmetry is the y-axis, the parabola opens down.<\/p>\n<\/div>\n<\/div>\n<p>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<p>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<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204543\/CNX_Precalc_Figure_10_03_0052.jpg\" width=\"487\" height=\"514\" alt=\"image\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.<\/h3>\n<ul>\n<li>Determine which of the standard forms applies to the given equation: [latex]{y}^{2}=4px[\/latex] or [latex]{x}^{2}=4py[\/latex].<\/li>\n<li>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<ul>\n<li>If the equation is in the form [latex]{y}^{2}=4px[\/latex], then\n<ul>\n<li>the axis of symmetry is the <em>x<\/em>-axis, [latex]y=0[\/latex]<\/li>\n<li>set [latex]4p[\/latex] equal to the coefficient of <em>x <\/em>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<li>use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(p,0\\right)[\/latex]<\/li>\n<li>use [latex]p[\/latex] to find the equation of the directrix, [latex]x=-p[\/latex]<\/li>\n<li>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<li>If the equation is in the form [latex]{x}^{2}=4py[\/latex], then\n<ul>\n<li>the axis of symmetry is the <em>y<\/em>-axis, [latex]x=0[\/latex]<\/li>\n<li>set [latex]4p[\/latex] equal to the coefficient of <em>y <\/em>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<li>use [latex]p[\/latex] to find the coordinates of the focus, [latex]\\left(0,p\\right)[\/latex]<\/li>\n<li>use [latex]p[\/latex] to find equation of the directrix, [latex]y=-p[\/latex]<\/li>\n<li>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<li>Plot the focus, directrix, and focal diameter, and draw a smooth curve to form the parabola.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Parabola with Vertex (0, 0) and the <em>x<\/em>-axis as the Axis of Symmetry<\/h3>\n<p>Graph [latex]{y}^{2}=24x[\/latex]. Identify and label the <strong>focus<\/strong>, <strong>directrix<\/strong>, and endpoints of the <strong>focal diameter<\/strong>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q885427\">Show Solution<\/span><\/p>\n<div id=\"q885427\" class=\"hidden-answer\" style=\"display: none\">\n<p>The standard form that applies to the given equation is [latex]{y}^{2}=4px[\/latex]. Thus, the axis of symmetry is the <em>x<\/em>-axis. It follows that:<\/p>\n<ul>\n<li>[latex]24=4p[\/latex], so [latex]p=6[\/latex]. Since [latex]p>0[\/latex], the parabola opens right\u00a0the coordinates of the focus are [latex]\\left(p,0\\right)=\\left(6,0\\right)[\/latex]<\/li>\n<li>the equation of the directrix is [latex]x=-p=-6[\/latex]<\/li>\n<li>the endpoints of the focal diameter\u00a0have the same <em>x<\/em>-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>\n<\/ul>\n<p>Next we plot the focus, directrix, and focal diameter, and draw a smooth curve to form the <strong>parabola<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204545\/CNX_Precalc_Figure_10_03_0192.jpg\" width=\"487\" height=\"376\" alt=\"image\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph [latex]{y}^{2}=-16x[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q277220\">Show Solution<\/span><\/p>\n<div id=\"q277220\" class=\"hidden-answer\" style=\"display: none\">\n<p>Focus: [latex]\\left(-4,0\\right)[\/latex]; Directrix: [latex]x=4[\/latex]; Endpoints of the latus rectum: [latex]\\left(-4,\\pm 8\\right)[\/latex]<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3276\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02180919\/leftopen.jpg\" alt=\"\" width=\"487\" height=\"366\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=23511&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"550\"><\/iframe><\/p>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Graphing a Parabola with Vertex (0, 0) and the <em>y<\/em>-axis as the Axis of Symmetry<\/h3>\n<p>Graph [latex]{x}^{2}=-6y[\/latex]. Identify and label the <strong>focus<\/strong>, <strong>directrix<\/strong>, and endpoints of the <strong>focal diameter<\/strong>.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q951555\">Show Solution<\/span><\/p>\n<div id=\"q951555\" class=\"hidden-answer\" style=\"display: none\">\n<p>The standard form that applies to the given equation is [latex]{x}^{2}=4py[\/latex]. Thus, the axis of symmetry is the <em>y<\/em>-axis. It follows that:<\/p>\n<ul>\n<li>[latex]-6=4p[\/latex], so [latex]p=-\\frac{3}{2}[\/latex]. Since [latex]p<0[\/latex], the parabola opens down.<\/li>\n<li>the coordinates of the focus are [latex]\\left(0,p\\right)=\\left(0,-\\frac{3}{2}\\right)[\/latex]<\/li>\n<li>the equation of the directrix is [latex]y=-p=\\frac{3}{2}[\/latex]<\/li>\n<li>the endpoints of the focal diameter 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>\n<\/ul>\n<p>Next we plot the focus, directrix, and latus rectum, and draw a smooth curve to form the <strong>parabola<\/strong>.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/11\/03204547\/CNX_Precalc_Figure_10_03_007n2.jpg\" width=\"487\" height=\"327\" alt=\"image\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Graph [latex]{x}^{2}=8y[\/latex]. Identify and label the focus, directrix, and endpoints of the focal diameter.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q824847\">Show Solution<\/span><\/p>\n<div id=\"q824847\" class=\"hidden-answer\" style=\"display: none\">\n<p>Focus: [latex]\\left(0,2\\right)[\/latex]; Directrix: [latex]y=-2[\/latex]; Endpoints of the latus rectum: [latex]\\left(\\pm 4,2\\right)[\/latex].<\/p>\n<p><a href=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\"><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-full wp-image-3277\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2017\/02\/02181513\/CNX_Precalc_Figure_10_03_0082.jpg\" alt=\"\" width=\"487\" height=\"365\" \/><\/a><\/p>\n<\/div>\n<\/div>\n<\/div>\n<h2>Writing Equations of Parabolas in Standard Form<\/h2>\n<p>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<div class=\"textbox\">\n<h3>How To: Given its focus and directrix, write the equation for a parabola in standard form.<\/h3>\n<ul>\n<li>Determine whether the axis of symmetry is the <em>x<\/em>&#8211; or <em>y<\/em>-axis.\n<ul>\n<li>If the given coordinates of the focus have the form [latex]\\left(p,0\\right)[\/latex], then the axis of symmetry is the <em>x<\/em>-axis. Use the standard form [latex]{y}^{2}=4px[\/latex].<\/li>\n<li>If the given coordinates of the focus have the form [latex]\\left(0,p\\right)[\/latex], then the axis of symmetry is the <em>y<\/em>-axis. Use the standard form [latex]{x}^{2}=4py[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Multiply [latex]4p[\/latex].<\/li>\n<li>Substitute the value from Step 2 into the equation determined in Step 1.<\/li>\n<\/ul>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Writing the Equation of a Parabola in Standard Form Given its Focus and Directrix<\/h3>\n<p>What is the equation for the <strong>parabola<\/strong> with <strong>focus<\/strong> [latex]\\left(-\\frac{1}{2},0\\right)[\/latex] and <strong>directrix<\/strong> [latex]x=\\frac{1}{2}?[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q259208\">Show Solution<\/span><\/p>\n<div id=\"q259208\" class=\"hidden-answer\" style=\"display: none\">\n<p>The focus has the form [latex]\\left(p,0\\right)[\/latex], so the equation will have the form [latex]{y}^{2}=4px[\/latex].<\/p>\n<p>Multiplying [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].<\/p>\n<p>Therefore, the equation for the parabola is [latex]{y}^{2}=-2x[\/latex].<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>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]?<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q886076\">Show Solution<\/span><\/p>\n<div id=\"q886076\" class=\"hidden-answer\" style=\"display: none\">\n<p>[latex]{x}^{2}=14y[\/latex]<\/p>\n<\/div>\n<\/div>\n<p><iframe loading=\"lazy\" id=\"mom2\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=29711&amp;theme=oea&amp;iframe_resize_id=mom2\" width=\"100%\" height=\"350\"><span style=\"width: 0px;overflow: hidden;line-height: 0\" class=\"mce_SELRES_start\">\ufeff<\/span><\/iframe><\/p>\n<\/div>\n<p>&nbsp;<\/p>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-2393\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li>Revision and Adaptation. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>College Algebra. <strong>Authored by<\/strong>: Abramson, Jay et al.. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2\">http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: Download for free at http:\/\/cnx.org\/contents\/9b08c294-057f-4201-9f48-5d6ad992740d@5.2<\/li><li>Question ID 23511. <strong>Authored by<\/strong>: Shahbazian,Roy, mb McClure,Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><li>Question ID 23711. <strong>Authored by<\/strong>: McClure,Caren. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em>. <strong>License Terms<\/strong>: IMathAS Community License CC-BY + GPL<\/li><\/ul><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Specific attribution<\/div><ul class=\"citation-list\"><li>Precalculus. <strong>Authored by<\/strong>: OpenStax College. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface\">http:\/\/cnx.org\/contents\/fd53eae1-fa23-47c7-bb1b-972349835c3c@5.175:1\/Preface<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":21,"menu_order":11,"template":"","meta":{"_candela_citation":"[{\"type\":\"cc-attribution\",\"description\":\"Precalculus\",\"author\":\"OpenStax 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