At (6, 2), 36 = 4p(2) so p = 4.5 thus the focus is at the point (0, 4.5)Īpplication The towers of a suspension bridge are 800 ft apart and rise 160 ft above the road. Since the parabola is vertical and has its vertex at (0, 0) its equation must be of the form: x2 = 4py The receiver should be placed 4.5 feet above the base of the dish. How far from the base should the receiver be placed? 12 2 (-6, 2) (6, 2) Consider a parabola cross-section of the dish and create a coordinate system where the origin is at the base of the dish. The dish is 12 ft in diameter and 2 ft deep. Ĭan you tell whether the parabola opens up or down by the location of the directrix? How?Īpplications A satellite dish is in the shape of a parabolic surface.
d1 Directrix The length of the latus rectum is |4p| where p is the distance from the vertex to the focus.Įquations of a Parabola y = ax2 + bx + c Vertical Hyperbola -b 2a Vertex: x = If a > 0, opens up If a 0, opens up If 4p < 0, opens down The directrix is horizontal and the vertex is midway between the focus and directrix Remember: |p| is the distance from the vertex to the focusĮX 1: Find the standard form of the equation of the parabola given: the vertex is (2, -3) and focus is (2, -5) Because of the location of the vertex and focus this must be a vertical parabola that opens down Equation: (x – h)2 = 4p(y – k) |p| = 2 V Equation: (x – 2)2 = -8(y + 3) F The vertex is midway between the focus and directrix, so the directrix for this parabola is y = -1Įx 2: find the standard form equation of the parabola if… the focus is (2,5) and the directrix is y =3 Because of the location of the vertex and focus this must be a vertical parabola that opens up F Equation: (x – h)2 = 4p(y – k) Vertex is the mid point between focus and directrix (2, 4) |p| = 1 Equation: (x – 2)2 = 4(y - 4) Foil and solve for y.Įx 3: find the standard form equation of the parabola if… the focus is (-1,-4) and the directrix is y =-6 Because of the location of the vertex and focus this must be a vertical parabola that opens up Equation: (x – h)2 = 4p(y – k) Vertex is the mid point between focus and directrix ( -1, -5) |p| = 1 Equation: (x +1)2 = 4(y +6) Foil and solve for y. For any point Q that is on the parabola, d2 = d1 Q d2 Focus The latus rectum of a parabola is a line segment that passes through the focus, is parallel to the directrix and has its endpoints on the parabola. Parabola A parabola is a set of points in a plane that are equidistant from a fixed line, the directrix, and a fixed point, the focus. For any point Q that is on the parabola, d2 = d1 Q d2 Focus Focus: a point on the axis of symmetry that is “p” units from the vertex d1 Directrix Directx: a line that is perpendicular to the axis of symmetry “p” units from the vertex.
Forms of equations y = a(x – h)2 + k opens up if a is positive opens down if a is negative vertex is (h, k) y = ax2 + bx + c opens up if a is positive V opens down if a is negative -b 2a -b 2a vertex is, f( ) Thus far in this course we have studied parabolas that are vertical - that is, they open up or down and the axis of symmetry is vertical Parabolas Things you should already know about a parabola. Unit 2: Day 8 Equation of a Parabola with Focus and DirectX
0 kommentar(er)
