Conic Sections

Parabola

Conic Sections - Parabola

The intersection of a plane with one nappe of the cone is a parabola.

Conic Sections - Parabola

The parabola has the characteristic shape shown above. A parabola is defined to be the “set of points the same distance from a point and a line”.

Conic Sections - Parabola

The line is called the directrix and the point is called the focus.

Focus

Directrix

Conic Sections - Parabola

The line perpendicular to the directrix passing through the focus is the axis of symmetry. The vertex is the point of intersection of the axis of symmetry with the parabola.

Focus

Directrix

Axis of Symmetry

Vertex

Conic Sections - Parabola

The definition of the parabola is the set of points the same distance from the focus and directrix. Therefore, d1 = d2 for any point (x, y) on the parabola.

Focus

Directrix

d1

d2

Latus Rectum

Parabola

Conic Sections - ParabolaThe latus rectum is the line segment passing through the focus, perpendicular to the axis of symmetry with endpoints on the parabola.

y = ax2

Focus

Vertex(0, 0)

LatusRectum

Example 1

Graph a parabola.

Find the vertex, axis of symetry, focus, directrix, and latus rectum.

Parabola – Example 1

Graph the function. Find the vertex, axis of symmetry, focus, directrix, and latus rectum

2)2()3(8 xy

Example 2

Graph a parabola using the vertex, focus, axis of symmetry and latus rectum.

Parabola – Example 2

Find the vertex, axis of symmetry, focus, directrix, endpoints of the latus rectum and sketch the graph.

(y 1)2 16(x 2)

Building a Table of Rules

Parabola

Building a Table of Rules

4p(y – k) = (x – h)2

p>0 opens up

p<0 opens down

Vertex: (h, k)

Focus: (h, k + p)

Directrix: y = k – p

Latus Rectum: |4p|

4p(x – h) = (y – k)2

p>0 opens right

p<0 opens left

Vertex: (h, k)

Focus: (h + p, k)

Directrix: x = h + p

Latus Rectum: |4p|

Paraboloid Revolution

Parabola

Paraboloid Revolution

A paraboloid revolution results from rotating a parabola around its axis of symmetry as shown at the right.

http://commons.wikimedia.org/wiki/Image:ParaboloidOfRevolution.pngGNU Free Documentation License

Paraboloid Revolution

They are commonly used today in satellite technology as well as lighting in motor vehicle headlights and flashlights.

Paraboloid Revolution

The focus becomes an important point. As waves approach a properly positioned parabolic reflector, they reflect back toward the focus. Since the distance traveled by all of the waves is the same, the wave is concentrated at the focus where the receiver is positioned.

Example 4 – Satellite Receiver

A satellite dish has a diameter of 8 feet. The depth of the dish is 1 foot at the center of the dish. Where should the receiver be placed?

8 ft

1 ft

Let the vertex be at (0, 0). What are the coordinates of a point at the diameter of the dish?

V(0, 0)

(?, ?)

Sample Problems

Parabola

Sample Problems

1. (y + 3)2 = 12(x -1)

a. Find the vertex, focus, directrix, and length of the latus rectum.

b. Sketch the graph.

c. Graph using a grapher.

Sample Problems

2. 2x2 + 8x – 3 + y = 0

a. Find the vertex, focus, directrix, axis of symmetry and length of the latus rectum.

b. Sketch the graph.

c. Graph using a grapher.

Sample Problems

3. Write the equation of a parabola with vertex at

(3, 2) and focus at (-1, 2).

Plot the known points.

What can be determined from these points?

Sample Problems

4. Write the equation of a parabola with focus at

(4, 0) and directrix y = 2.

Graph the known values.

What can be determined from the graph?

The parabola opens down and has a model of4p(y + k) = (x – h)2

What is the vertex?

Sample Problems

4. Write the equation of a parabola with focus at

(4, 0) and directrix y = 2.

The vertex must be on the axis of symmetry, the same distance from the focus and directrix. The vertex must be the midpoint of the focus and the intersection of the axis and directrix.

The vertex is (4, 1)

Parabola – Assignment

Wksheet #12-15**, 28-31, 36, 37,

40-43

**Find the vertex, direction of opening, focus, directrix, latus rectum.

Please do your assignment on graph paper!!