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.

circle ellipse parabola hyperbola

Degenerate Conic SectionsAtypical 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: Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, where A, B, and C are not all zero

point line intersecting lines

ParabolasDefinition: 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

directrixvertex

axis

By definition, the distance between F and P has to equal the distance between P and D.

x2 +y2 – 2py + p2 = y2 + 2py +p2

x2 = 4py

The standard form of the equation of a parabola that opens upward or downward

2 2( 0) ( ) distance between F &Px y p

F(0,p)

P(x,y)

D(x,-p)2 2( ) ( ( )) distance between P & Dx x y p

2 2 2 2( 0) ( ) ( - ) ( ( ))x y p x x y 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 y2 = 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 Equation x2=4py y2 = 4px

Opens Up if p>0

Down if p<0

To right if p>0

To left if p<0

Focus (0, p) (p, 0)

Directrix y = -p x = -p

Axis y-axis x-axis

Focal length p p

Focal width |4p| |4p|

Example 1aFind the focus, the directrix, and focal width of the parabola

21

12y x

Example 1bFind the focus, the directrix, and focal width of the parabola

x = 2y2

Example 2Find 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 3Find 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

HomeworkPage 641: 1,2, 5-20