Presentation on theme: "Today in Precalculus Notes: Conic Sections - Parabolas Homework"— Presentation transcript:
1Today in Precalculus Notes: Conic Sections - Parabolas Homework Test grades will be in home access before I leave today. We’ll go over them on Monday.
2Conic SectionsConic sections are formed by the intersection of a plane and a cone.hyperbolacircleellipseparabola
3Degenerate Conic Sections Atypical conicsThe conic sections can be defined algebraically in the Cartesian plane as the graphs of second-degree equations in two variables, that is, equations of the form: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, and C are not all zeropointlineintersecting lines
4ParabolasDefinition: A parabola is the set of all points in a plane equidistant from the directrix and the focus in the plane.The line passing through the focus and perpendicular to the directrix is the axis of the parabola and is the line of symmetry for the parabola.The vertex is midway between the focus and the directrix and is the point of the parabola closest to both.axisfocusvertexdirectrix
5P(x,y)By definition, the distance between F and P has to equal the distance between P and D.x2 +y2 – 2py + p2 = y2 + 2py +p2x2 = 4pyThe standard form of the equation of a parabola that opens upward or downwardF(0,p)D(x,-p)
6If p>0, the parabola opens upward, if p<0 it opens downward. Parabolas that open to the left or right are inverse relations of upward or downward opening parabolas.So equations of parabolas with vertex (0,0) that open to the right or to the left have the standard form y2 = 4pxIf p>0, the parabola opens to the right and if p<0, the parabola opens to the left.
7p is the focal length of the parabola – the directed distance from the vertex to the focus of the parabola.A line segment with endpoints on a parabola is a chord of the parabola.The value |4p| is the focal width of the parabola – the length of the chord through the focus and perpendicular to the axis.
8Parabolas with vertex (0,0) Standard Equationx2=4pyy2 = 4pxOpensUp if p>0Down if p<0To right if p>0To left if p<0Focus(0, p)(p, 0)Directrixy = -px = -pAxisy-axisx-axisFocal lengthpFocal width|4p|
9Example 1aFind the focus, the directrix, and focal width of the parabola
10Example 1bFind the focus, the directrix, and focal width of the parabolax = 2y2
11Example 2 Find the equation in standard form for a parabola whose directrix is the line x = 5 and focus is the point (-5, 0)b) directrix is the line y =6 and vertex is (0,0)
12Example 3 Find the equation in standard form for a parabola whose vertex is (0,0) and focus is (0, -4)b) vertex is (0,0), opens to the left with focal width7