Warm UpNO CALCULATOR 1) Determine the equation for the graph shown. 2)Convert the equation from polar to rectangular. r = 3cosθ + 2sin θ 3)Convert the equation from rectangular to polar. (x + 2) 2 + y 2 = 4

Polar Graphs Homework ANSWERS

Homework Answers

Write the equation, focus and directrix of a parabola

Conic Sections A conic section (or conic) is a cross section of a cone – the intersection of a plane with a right circular cone. The 3 basic conic sections are the parabola, ellipse and hyperbola. (circle is a special ellipse)

Parabolas A parabola is the set of all points in a plane equidistant from a particular line (the directrix) and a particular point (the focus)

The standard (vertex) form equation of a parabola with a vertex at (h, k) and where p represents the directed distance between the focus and vertex (called the focal length). Equation of a Parabola

Identify the direction of the opening y – 3 = -5(x+1) 2 y 2 = -2x x = -y 2 + 3y 1- 2y + x 2 = 0

1. Write an equation of the parabola with vertex (2, 1) and focus (2, 4) 2. Write an equation of the parabola that passes through the point (2, 0) with a vertical axis of symmetry passing through the vertex (3, 1). Examples

3. Write an equation of the parabola with focus (2, -3) and directrix x = 8 Examples (cont.)

the focal width of a parabola is the length of the vertical (or horizontal) line segment that passes through the focus and touches the parabola at each end. |4p| is the focal width.

Identify the Parts a) Vertex: b) Opening: c) Axis of Symmetry d) Focal length: e) Directrix: f) Focus: g) Focal width:

Identify the Parts a) Vertex: b) Opening: c) Axis of Symmetry d) Focal length: e) Directrix: f) Focus: g) Focal width:

Completing the Square First, decide which way your parabola opens(up, down, right or left)! Is it x = or y = ? Example: 24x = 4x 2 – y + 1

EX: y = 4x 2 – 8x + 3 a) Vertex form: b) Vertex: c) Opening: d) Focal length: e) Directrix: f) Focus: g) Focal width: Parts of a Parabola (cont.)

EX: y 2 + 6y + 8x + 25 = 0 a) Vertex form: b) Vertex: c) Opening: d) Focal length: e) Directrix: f) Focus: g) Focal width: Parts of a Parabola (cont.)

Applications of parabolas

A signal light on a ship is a spotlight with parallel reflected light rays (see the figure). Suppose the parabolic reflector is 12 inches in diameter and 6 inches deep. How far from the vertex should the light source be placed so that the beams of light will run parallel to the axis of its mirror?

Homework change Omit #s 5 – 6 No proof on #s