{"id":1914,"date":"2015-11-12T18:30:43","date_gmt":"2015-11-12T18:30:43","guid":{"rendered":"https:\/\/courses.candelalearning.com\/collegealgebra1xmaster\/?post_type=chapter&p=1914"},"modified":"2015-11-12T18:30:43","modified_gmt":"2015-11-12T18:30:43","slug":"graphing-parabolas-with-vertices-at-the-origin","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ivytech-collegealgebra\/chapter\/graphing-parabolas-with-vertices-at-the-origin\/","title":{"raw":"Graphing Parabolas with Vertices at the Origin","rendered":"Graphing Parabolas with Vertices at the Origin"},"content":{"raw":"

In The Ellipse<\/a>, we saw that an 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 parabola<\/strong>.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 2.<\/b> Parabola[\/caption]\n\nLike the ellipse and 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 directrix<\/strong>, and a fixed point (the focus<\/strong>) not on the directrix.\n\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. See Figure 3. 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.\n\nThe line segment that passes through the focus and is parallel to the directrix is called the latus rectum<\/strong>. The endpoints of the latus rectum lie 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.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 3.<\/b> Key features of the parabola[\/caption]\n\nTo work with parabolas in the coordinate plane<\/strong>, we consider two cases: those with a vertex at the origin and those with a vertex<\/strong> at a point other than the origin. We begin with the former.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 4<\/b>[\/caption]\n\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 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 distance formula<\/strong>.\n<\/p>

[latex]\\begin{array}{l}d=\\sqrt{{\\left(x - 0\\right)}^{2}+{\\left(y-p\\right)}^{2}}\\hfill \\\\ =\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}\\hfill \\end{array}[\/latex]<\/div>\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].\n
[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]<\/div>\nWe then square both sides of the equation, expand the squared terms, and simplify by combining like terms.\n
[latex]\\begin{array}{c}{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\\\\ \\text{ }{x}^{2}=4py\\end{array}[\/latex]<\/div>\nThe equations of parabolas with vertex [latex]\\left(0,0\\right)[\/latex] are [latex]{y}^{2}=4px[\/latex] when the x<\/em>-axis is the axis of symmetry and [latex]{x}^{2}=4py[\/latex] when the y<\/em>-axis is the axis of symmetry. These standard forms are given below, along with their general graphs and key features.\n
\n

A General Note: Standard Forms of Parabolas with Vertex (0, 0)<\/h3>\nThe table below\u00a0and Figure 5\u00a0summarize the standard features of parabolas with a vertex at the origin.\n
Axis of Symmetry<\/strong><\/td>\nEquation<\/strong><\/td>\nFocus<\/strong><\/td>\nDirectrix<\/strong><\/td>\nEndpoints of Latus Rectum<\/strong><\/td>\n<\/tr>
x<\/em>-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>
y<\/em>-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><\/tbody><\/table>\n[caption id=\"\" align=\"aligncenter\" width=\"975\"]\"\"Figure 5.<\/b> (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]<\/div>\nThe key features of a parabola are its vertex, axis of symmetry, focus, directrix, and latus rectum. When given a standard equation for a parabola centered at the origin, we can easily identify the key features to graph the parabola.\n\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 latus rectum, these lines intersect on the axis of symmetry, as shown in Figure 6.\n\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\"Figure 6<\/b>[\/caption]\n\n
\n

How To: Given a standard form equation for a parabola centered at (0, 0), sketch the graph.<\/h3>\n