Section 10-2Pages 735-743

Introduction to Conics:

Parabolas

Objectives

• I can write equations for parabolas in Standard Format

• I can graph parabolas by finding the key information for each

• I can complete the square to obtain vertex format

How to identify types of Conic Sections from General Form

PARABOLAS CIRCLES

Either x or y is x and y are both

squared but not squared with the

both. same coefficient.

ELLIPSES HYPERBOLASx and y are both x and y are both

squared with different squared, 1 is positive

coefficients but the 1 is negative.

same signs.

Conic Sections

A conic section is the intersection of a plane and a double cone.

Conical View

Parabolas (10-2)

A Parabola is the set of all points in a plane that are equidistant between a fixed point (focus) and a line (directrix).

Real Parabolas

• Flashlights• Headlights• Mirrors • Projectile Motion• Satellite Dish• Architecture

Package

A Little Review

• You know the basic equation of any parabola as• y = ax2 + bx + c• You can write this in vertex format and it

becomes: y = a(x – h)2 + k• In that format, the vertex is (h, k); and the axis of

symmetry is x = h• In addition, if a > 0, then the parabola opened

upwards, if a < 0 then the parabola opened down.

Review: Complete the Square• Write y = 2x2 + 12x + 14 in vertex format• y = 2x2 + 12x + 14 (Underline variables)• y = 2 (x2 + 6x) + 14 (Factor out the 2)• y = 2 (x2 + 6x + _____) + 14 - _______• y = 2 (x2 + 6x + 9) + 14 – 18• y = 2 (x + 3)(x + 3) - 4• y = 2 (x + 3)2 – 4• Vertex (-3, -4)• Axis of Sym: x = -3 ;• Opens: Upward since a > 0

A New Review

• The new equation we look at is x = ay2 + by + c• The new basic equation of a parabola is

x = a(y – k)2 + h• In that format, the vertex is (h, k); and the axis of

symmetry is y = k• In addition, if a > 0, then the parabola opened

Right, if a < 0 then the parabola opened Left.

Parabola

• A parabola is a set of points in a plane that are all the same distance from a fixed line called the directrix and a fixed point not on the line called the focus .

Vocabulary

• Any line segment that passes through the focus point with endpoints on the parabola is called a focal chord

• The perpendicular chord to the AOS is called the latus rectum(LR)

Latus rectum

1LR

a

Key Concept

• The distance from Vertex Point to Focus Point is “p”

• This is also the same distance from Vertex Point to the Directrix Line

ap

4

1

Parabolas

Where to find them…• FOCUS: inside the parabola

• DIRECTRIX: outside of the parabola

• AOS: through the vertex, perpendicular to the directrix

• FOCAL CHORD (latus rectum): inside of the parabola

When graphing you MUST label…

• vertex

• focus

• directrix

• axis of symmetry

• endpoints of the focal chord

21( 2) 3

8x y : ( 3, 2)Vertex

Opens Left

12 units

4a

: ( 5, 2)Focus

Directrix: -1x

8 unitsLR

Homework

• WS 10-1