Conic Sections and

ParabolasChapter 8.1

Chapter 8.1• The study of conic sections, or simply conics.• Conics are formed by the intersection of a plane

with a pair of cones.

ParabolasAlgebraically speaking• y = ax2 + bx + c• Has a U-shape.• Opens with upward or downward.• The lowest or highest point on such a parabola is

called the vertex.• It is symmetric about the its axis called the axis

of symmetry.

ParabolasGeometrically speaking• The set of all points in the plane equidistant from

a fixed point F, called the focus and a fixed line , called the directrix.

• The vertex V, lies halfway between the focus and the directrix.

• The axis of runs through the focus and the vertex.• The axis of symmetry is perpendicular to the

directrix.

Parabolas• If x and y is any point on the parabola, then the

distance from (x, y) to the focus F (0, p) is or • The distance from (x, y) to the directrix is or • = • x2 = 4py, General Form• If p > 0, the parabola opens upward• If p < 0, the parabola opens downward.• It has a vertical axis of symmetry

Standard Form• The Standard Form of the equation of a

parabola with vertex at (h, k) is:• (x - h)2 = 4p (y- k), vertical axis; directrix at y = k

- p • (y - k)2 = 4p(x - h), horizontal axis; directrix at x =

h - p• The focus lies on the axis p units from the vertex.

• Find the standard form of the equation of the parabola with vertex (2, 1) and focus (2, 4)

Standard Form• Find the focus of the parabola

Standard Form• Find the standard form of the parabola with

vertex at the origin and focus (2, 0)

Parabola• The line segment that runs through the focus

perpendicular to the axis, with endpoints on the parabola is called the latus rectum.

• Its length is the focal diameter of the parabola.

• Find the focus, directrix, and focal diameter of the parabolas , and sketch the graph.