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10.1 Parabolas.notebook
February 25, 2015
Do Now: Write the equation in standard form. Find the center and the radius. Sketch the graph.
3x2 + 3y2 ­ 12x + 18y = 9
Definition: A parabola is the set of all points in a plane equidistant
from a fixed point F (the focus) and a fixed line l (the directrix) that
lie in the plane.
Vertex: midpoint between focus and directrix
on the axis (the point closest to directrix)
Axis: line passing through the focus and vertex
(perpendicular to the directrix)
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10.1 Parabolas.notebook
February 25, 2015
To get a formula for a parabola...
place the vertex on the origin and call the focus (0, p)
We know the distance between P & F and the distance between P & P' must be equal!
Standard form with vertex (0, 0):
Vertical Axis:
Horizontal Axis:
where
p≠0
where
p≠0
focus: (0, p)
focus: (p, 0)
directrix: y = ­p
directrix: x = ­p
if p > 0
if p > 0
if p < 0
if p < 0
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10.1 Parabolas.notebook
February 25, 2015
Example:
Find vertex, focus, directrix, and axis of symmetry. Sketch the graph. 1). y = (1/6)x2
2). y2 = 2x
V:
V:
F:
F:
d:
d:
axis:
axis:
Example:
(a) Find an equation of a parabola that has vertex
at the origin, opens right, and passes through P(7, ­3).
(b) Find the focus of the parabola.
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10.1 Parabolas.notebook
February 25, 2015
Translations: Standard position with vertex (h, k)
­distance from vertex to focus is p
shift right/left
Vertical Axis:
1
2
shift
up/down
Horizontal Axis: 1
2
Find vertex, focus, axis, and directrix. Sketch. 3. (x + 1/2)2 = ­4(y ­ 3)
V:
F:
d:
axis:
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10.1 Parabolas.notebook
February 25, 2015
4. y2 + 8y ­ 12x + 4 = 0
V:
F:
d:
axis:
Find an equation of the parabola.
5. Vertex: (0, 0)
Focus: (0, ­4)
7. V: (­1, 2) F: (­1, 0)
6. Vertex: (0, 0)
Directrix: x = 3
8. V: (­2, 1) d: x = 1
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10.1 Parabolas.notebook
February 25, 2015
Find the tangent line at a point on a parabola
9. Find the equation of the tangent line to the parabola given by y = x2 at the point (1, 1).
10. Find the equation of the tangent line to the parabola given by y = ­2x2 at the point (2,­8).
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10.1 Parabolas.notebook
February 25, 2015
Homework:
10.1 pg. 701 ­ 703 #1 ­ 23 odd, 27 ­ 59 odd, 65, 66
10.3 pg. 721 # 45 ­ 51 odd
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