{"id":429,"date":"2019-07-15T22:44:57","date_gmt":"2019-07-15T22:44:57","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/waymakercollegealgebracorequisite\/chapter\/summary-the-parabola\/"},"modified":"2019-07-15T22:44:57","modified_gmt":"2019-07-15T22:44:57","slug":"summary-the-parabola","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/ntcc-collegealgebracorequisite\/chapter\/summary-the-parabola\/","title":{"raw":"Summary: The Parabola","rendered":"Summary: The Parabola"},"content":{"raw":"\n
Key Equations<\/h2>\n
\n\n
\n
Parabola, vertex at origin, axis of symmetry on x<\/em>-axis<\/td>\n
[latex]{y}^{2}=4px[\/latex]<\/td>\n<\/tr>\n
\n
Parabola, vertex at origin, axis of symmetry on y<\/em>-axis<\/td>\n
[latex]{x}^{2}=4py[\/latex]<\/td>\n<\/tr>\n
\n
Parabola, vertex at [latex]\\left(h,k\\right)[\/latex], axis of symmetry on x<\/em>-axis<\/td>\n
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, and a fixed point (the focus) not on the directrix.<\/li>\n \t
The standard form of a parabola with vertex [latex]\\left(0,0\\right)[\/latex] and the x<\/em>-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\n \t
The standard form of a parabola with vertex [latex]\\left(0,0\\right)[\/latex] and the y<\/em>-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\n \t
When given the focus and directrix of a parabola, we can write its equation in standard form.<\/li>\n \t
The standard form of a parabola with vertex [latex]\\left(h,k\\right)[\/latex] and axis of symmetry parallel to the x<\/em>-axis can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\n \t
The standard form of a parabola with vertex [latex]\\left(h,k\\right)[\/latex] and axis of symmetry parallel to the y<\/em>-axis can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\n \t
Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides.<\/li>\n<\/ul>\n
Glossary<\/h2>\ndirectrix<\/strong> a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant\n\nfocus (of a parabola)<\/strong> a fixed point in the interior of a parabola that lies on the axis of symmetry\n\nfocal diameter (latus rectum)<\/strong> the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola\n\nparabola<\/strong> 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, and a fixed point (the focus) not on the directrix\n","rendered":"
Key Equations<\/h2>\n
\n\n
\n
Parabola, vertex at origin, axis of symmetry on x<\/em>-axis<\/td>\n
[latex]{y}^{2}=4px[\/latex]<\/td>\n<\/tr>\n
\n
Parabola, vertex at origin, axis of symmetry on y<\/em>-axis<\/td>\n
[latex]{x}^{2}=4py[\/latex]<\/td>\n<\/tr>\n
\n
Parabola, vertex at [latex]\\left(h,k\\right)[\/latex], axis of symmetry on x<\/em>-axis<\/td>\n
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, and a fixed point (the focus) not on the directrix.<\/li>\n
The standard form of a parabola with vertex [latex]\\left(0,0\\right)[\/latex] and the x<\/em>-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\n
The standard form of a parabola with vertex [latex]\\left(0,0\\right)[\/latex] and the y<\/em>-axis as its axis of symmetry can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\n
When given the focus and directrix of a parabola, we can write its equation in standard form.<\/li>\n
The standard form of a parabola with vertex [latex]\\left(h,k\\right)[\/latex] and axis of symmetry parallel to the x<\/em>-axis can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens right. If [latex]p<0[\/latex], the parabola opens left.<\/li>\n
The standard form of a parabola with vertex [latex]\\left(h,k\\right)[\/latex] and axis of symmetry parallel to the y<\/em>-axis can be used to graph the parabola. If [latex]p>0[\/latex], the parabola opens up. If [latex]p<0[\/latex], the parabola opens down.<\/li>\n
Real-world situations can be modeled using the standard equations of parabolas. For instance, given the diameter and focus of a cross-section of a parabolic reflector, we can find an equation that models its sides.<\/li>\n<\/ul>\n
Glossary<\/h2>\n
directrix<\/strong> a line perpendicular to the axis of symmetry of a parabola; a line such that the ratio of the distance between the points on the conic and the focus to the distance to the directrix is constant<\/p>\n
focus (of a parabola)<\/strong> a fixed point in the interior of a parabola that lies on the axis of symmetry<\/p>\n
focal diameter (latus rectum)<\/strong> the line segment that passes through the focus of a parabola parallel to the directrix, with endpoints on the parabola<\/p>\n
parabola<\/strong> 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, and a fixed point (the focus) not on the directrix<\/p>\n\n\t\t\t \n\t\t\t