Warm UpNO CALCULATOR

Warm Up NO CALCULATOR1) Determine the equation for the graph shown.

(A) r 3 2cos(B) r 3 3sin(C) r 1 2sin(D) r 3sin(2 )(E) r 3sin

2) Convert the equation from polar to rectangular. r = 3cosθ + 2sin θ

3) Convert the equation from rectangular to polar. (x + 2)2 + y2 = 4

Polar Graphs Homework ANSWERS1) r 5 2) r 4 4cos

3) r 5sin4

Polar Graphs Homework ANSWERS

ParabolasWrite the equation, focus and directrix of a

parabola

Conic SectionsA 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)

ParabolasA 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

21y k (x h)4p 21x h (y k)4p

Identify the direction of the opening

y – 3 = -5(x+1)2

y2 = -2x

x = -y2 + 3y

1- 2y + x2 = 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 Parts21y 3 (x 1)8

a) Vertex:

b) Opening:

c) Axis of Symmetry

d) Focal length:

e) Directrix:

f) Focus:

g) Focal width:

Identify the Parts2x 2 2(y 1)

a) Vertex:

b) Opening:

c) Axis of Symmetry

d) Focal length:

e) Directrix:

f) Focus:

g) Focal width:

Completing the SquareFirst, decide which way your parabola opens(up, down, right or left)! Is it x = or y = ?Example:24x = 4x2 – y + 1

EX: y = 4x2 – 8x + 3a) Vertex form:

b) Vertex:

c) Opening:

d) Focal length:

e) Directrix:

f) Focus:

g) Focal width:

Parts of a Parabola (cont.)

EX: y2 + 6y + 8x + 25 = 0a) 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 parallelreflected light rays (see the figure). Suppose the parabolicreflector 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?