Presentation on theme: "Today 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."— Presentation transcript:
Today 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.
Conic Sections Conic sections are formed by the intersection of a plane and a cone. circleellipse parabola hyperbola
Degenerate Conic Sections Atypical conics The 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: Ax 2 + Bxy + Cy 2 + Dx + Ey + F = 0, where A, B, and C are not all zero pointline intersecting lines
Parabolas Definition: 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. focus directrix vertex axis
By definition, the distance between F and P has to equal the distance between P and D. x 2 +y 2 – 2py + p 2 = y 2 + 2py +p 2 x 2 = 4py The standard form of the equation of a parabola that opens upward or downward F(0,p) P(x,y) D(x,-p)
If 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 y 2 = 4px If p>0, the parabola opens to the right and if p<0, the parabola opens to the left.
p 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.
Parabolas with vertex (0,0) Standard Equationx 2 =4pyy 2 = 4px OpensUp if p>0 Down if p<0 To right if p>0 To left if p<0 Focus(0, p)(p, 0) Directrixy = -px = -p Axisy-axisx-axis Focal lengthpp Focal width |4p|
Example 1a Find the focus, the directrix, and focal width of the parabola
Example 1b Find the focus, the directrix, and focal width of the parabola x = 2y 2
Example 2 Find the equation in standard form for a parabola whose a)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)
Example 3 Find the equation in standard form for a parabola whose a)vertex is (0,0) and focus is (0, -4) b) vertex is (0,0), opens to the left with focal width7