Section 9.3

The Parabola

Definition of a Parabola

Example

2

Find the vertex and the axis of symmetry of the parabola given

by y 2( 3) 4. Does it open up or down?x= − −

Standard Form of the Equation of a Parabola

Example

2

Find the focus and the directrix of the parabola given

by y 24 .x=

Using the Standard Form of the Equation of a Parabola

Example

2

Find the focus and the directrix of the parabola

given by x 4 .y= −

Example

2

Find the focus and the directrix of the parabola

given by y 12 .x= −

Translations of Parabola

Example

Find the standard form of the equation of a parabola

with focus (0,3) and directrix of y=-1

Example

2

Find the vertex, focus and directrix of the parabola

given by (x-3) 8( 1).y= +

Example

2

Find the vertex, axis, focus, and directrix of the parabola

by completing the square on the equation

8y=2x 12 14.x− −

Example

2

Find the vertex, axis, focus, and directrix of the parabola

by completing the square on the equation

6 16 25 0.y y x− + + =

Applications

Applications of Parabolas- cables on suspension bridges, arches on bridges, solar

cookers, reflectors on lights, satellite dishes

Example

A student scientist wants to construct a solar cooker which is

in the shape of a parabolic dish with a diameter of 6 feet and a

depth of 1 foot. Where should the cooker (focus) be located

to get the maximum reflected sun rays to cook the food.

6 feet

1 foot

(3,1)

(0,0)

cooker

Degenerate Conic Sections

Intersections might now result in a conic section. Three degenerate cases occur

when the cutting plane passes through the

vertex. These degenerate conic sections are a point, line, and a pair of intersecting lines.

(a)

(b)

(c)

(d)

2

Find the focus and directrix of the parabola with the given

equation. y 16 .x=

( 4,0), 4

(4,0), 4

(4,0), 4

(0, 4), 4

y

x

y

x

− = −

= −

=

=

(a)

(b)

(c)

(d)

2

Find the vertex of the following parabola

(x+4) 12( 2).y= +

(2,4)

(4,2)

( 4, 2)

( 2, 4)

− −

− −

(a)

(b)

(c)

(d)

2

Find the focus of the following parabola

(x+4) 12( 2).y= +

(2, 4)

(4,1)

( 4, 2)

( 4,1)

−

− −

−