{"id":5240,"date":"2018-05-10T21:09:01","date_gmt":"2018-05-10T21:09:01","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/coreq-collegealgebra\/?post_type=chapter&p=5240"},"modified":"2018-05-21T19:35:53","modified_gmt":"2018-05-21T19:35:53","slug":"the-parabola","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/odessa-coreq-collegealgebra\/chapter\/the-parabola\/","title":{"raw":"The Parabola*","rendered":"The Parabola*"},"content":{"raw":"Did you know that the Olympic torch is lit several months before the start of the games? The ceremonial method for lighting the flame is the same as in ancient times. The ceremony takes place at the Temple of Hera in Olympia, Greece, and is rooted in Greek mythology, paying tribute to Prometheus, who stole fire from Zeus to give to all humans. One of eleven acting priestesses places the torch at the focus of a parabolic mirror, which focuses light rays from the sun to ignite the flame.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"Description The Olympic torch concludes its journey around the world when it is used to light the Olympic cauldron during the opening ceremony. (credit: Ken Hackman, U.S. Air Force)[\/caption]\r\n\r\nParabolic mirrors (or reflectors) are able to capture energy and focus it to a single point. The advantages of this property are evidenced by the vast list of parabolic objects we use every day: satellite dishes, suspension bridges, telescopes, microphones, spotlights, and car headlights, to name a few. Parabolic reflectors are also used in alternative energy devices, such as solar cookers and water heaters, because they are inexpensive to manufacture and need little maintenance. In this section we will explore the parabola and its uses, including low-cost, energy-efficient solar designs.\r\n

Parabolas with Vertices at the Origin<\/h2>\r\nIn The Ellipse, 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>.\r\n\r\n[caption id=\"\" align=\"aligncenter\" width=\"487\"]\"\" Parabola[\/caption]\r\n\r\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.\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 latus rectum,\u00a0<\/strong>also called the\u00a0focal 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\"]\"\" Key features of the parabola[\/caption]\r\n\r\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.\r\n\r\n\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 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>.\r\n

[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]<\/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[latex]\\sqrt{{x}^{2}+{\\left(y-p\\right)}^{2}}=y+p[\/latex]\r\n\r\nWe then square both sides of the equation, expand the squared terms, and simplify by combining like terms.\r\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]<\/p>\r\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.\r\n

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