10.2 Parabolas10.2 Parabolas10.2 Parabolas10.2 Parabolas•Where is the focus and directrix compared to the Where is the focus and directrix compared to the vertex?vertex?•How do you know what direction a parabola opens?How do you know what direction a parabola opens?•How do you write the equation of a parabola given How do you write the equation of a parabola given the focus/directrix?the focus/directrix?•What is the general equation for a parabola?What is the general equation for a parabola?
Parabolas
focus
axis of symmetry
directrix
A parabola is defined in terms of a fixed point, called the focus, and a fixed line, called the directrix.
A parabola is the set of all points P(x,y) in the plane whose distance to the focusequals its distance to the directrix.
Horizontal Directrix
p > 0: opens upward
focus: (0, p)
directrix: y = –p
axis of symmetry: y-axisx
y
D(x, –p)
P(x, y)
F(0, p)
y = –pO
p < 0: opens downward
Standard Equation of a parabola with its vertex at the origin is
x2 = 4py
Vertical Directrix
p > 0: opens right
focus: (p, 0)
directrix: x = –p
axis of symmetry: x-axis
p < 0: opens left
Standard Equation of a parabola with its vertex at the origin is
x
y
D(x, –p)P(x, y)
F(p, 0)
x = –p
O
y2= 4px
Example 1Graph . Label the vertex, focus, and directrix. y2 = 4px
21x y
4
-4 -2
2
42
4
-4
-2
Identify p.
So, p = 1
Since p > 0, the parabola opens to the right.
Vertex: (0,0)Focus: (1,0)Directrix: x = -1
y2 = 4(1)x
Example 1Graph . Label the vertex, focus, and directrix. Y2 = 4x
21x y
4
-4 -2
2
42
4
-4
-2
Use a table to sketch a graph
y
x0
0
2
1
4
4
-2
1
-4
4
Example 2Write the standard equation of the parabola with its vertex at the origin and the directrix y = -6.
Since the directrix is below the vertex, the parabola opens upSince y = -p and y = -6,p = 6
x2=4(6)y x2 = 24y
• Where is the focus and directrix compared to vertex?
The focus is a point on the line of symmetry and the directrix is a line below the vertex. The focus and directrix are equidistance from the vertex.
• How do you know what direction a parabola opens?x2, graph opens up or down, y2, graph opens right or
left• How do you write the equation of a parabola given
the focus/directrix?Find the distance from the focus/directrix to the
vertex (p value) and substitute into the equation.• What is the general equation for a parabola?x2= 4py (opens up [p>0] or down [p<0]), y2 = 4px
(opens right [p>0] or left [p<0])
Assignmentp. 598, 16-21, 23-53 odd
10.2 Parabolas, day 2
• What does it mean if a parabola has a translated vertex?
• What general equations can you use for a parabola when the vertex has been translated?
Standard Equation of a Translated Parabola
Vertical axis:
vertex: (h, k)
focus: (h, k + p)
directrix: y = k – p
axis of symmetry: x = h
(x − h)2 = 4p(y − k)
Standard Equation of a Translated Parabola
Horizontal axis:
vertex: (h, k)
focus: (h + p, k)
directrix: x = h - p
axis of symmetry: y = k
(y − k)2 = 4p(x − h)
Example 3Write the standard equation of the parabola with a focus at F(-3,2) and directrix y = 4. Sketch the info.
The parabola opens downward, so the equation is of the form
vertex: (-3,3)
h = -3, k = 3
p = -1
(x − h)2 = 4p(y − k)
(x + 3)2 = 4(−1)(y − 3)
Example 4Write an equation of a parabola whose vertex is
at (−2,1) and whose focus is at (−3, 1).
Begin by sketching the parabola. Because the parabola opens to the left, it has the form
(y −k)2 = 4p(x − h)
Find h and k: The vertex is at (−2,1) so h = −2 and k = 1
Find p: The distance between the vertex (−2,1) and the focus (−3,1) by using the distance formula.
p = −1 (y − 1)2 = −4(x + 2)
• What does it mean if a parabola has a translated vertex?
It means that the vertex of the parabola has been moved from (0,0) to (h,k).
• What general equations can you use for a parabola when the vertex has been translated?
(y-k)2 =4p(x-h) (x-h)2 =4p(y-k)
Assignment
p. 598, 38-44 even, 54-68 even
p. 628, 15-16, 22, 28
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