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, d 1 = d 2 for any point (x, y) on the parabola. Focus Directrix d1d1 d2d2

Latus Rectum Parabola

Conic Sections - Parabola The latus rectum is the line segment passing through the focus, perpendicular to the axis of symmetry with endpoints on the parabola. y = ax 2 Focus Vertex (0, 0) Latus Rectum

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

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.

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. GNU 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 2.2x 2 + 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 of 4p(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, **Find the vertex, direction of opening, focus, directrix, latus rectum. Please do your assignment on graph paper!!