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Warm UpNO CALCULATOR. 1) Determine the equation for the graph shown . Convert the equation from polar to rectangular. r = 3cos θ + 2sin θ Convert the equation from rectangular to polar. (x + 2) 2 + y 2 = 4. Polar Graphs Homework ANSWERS. Polar Graphs Homework ANSWERS. Parabolas. - PowerPoint PPT PresentationTRANSCRIPT
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?