10.2 Parabolas 10.2 Parabolas Where is the focus and directrix compared to the vertex?Where is the...
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Transcript of 10.2 Parabolas 10.2 Parabolas Where is the focus and directrix compared to the vertex?Where is the...
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