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Precalculus – Chapter 8 – Parabolas

*Definition:*

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

Circle Ellipse Parabola Hyperbola

By changing the angle and location of intersection, we can produce a circle, ellipse, parabola or hyperbola; or in the special case when the plane touches the vertex: a point, line or 2 intersecting lines.

Point Line 2 Intersecting Lines

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.

**Parabola **

1

*p*

*0 x*

*2 p*

0

*4 py p*

*0 y*

2

*4 px p*

0

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.

2

Parabolas with Vertex (0, 0)

Standard Equation

Opens

Focus

Directrix

Axis

Focal length

*Focal width x*

2

*4 py y*

2

*4 px*

*Examples *

*Find the focus, the directrix, and focal width of the parabolas (a) y*

*(1/12) x*

*2 and (b) x*

*2 y*

2 .

(a) (b)

*Find an equation in standard form for the parabola whose (a) directrix is the line x = 5 and focus is the point (-5, 0) and (b) directrix is the line y*

6 and vertex is the point (0, 0) .

(a) (b)

Find an equation in standard form for the parabola whose (a) vertex is (0,0) and focus is (0, -4) and (b) vertex is (0,0), opens to the left with focal width 7

(a) (b)

3