9.6 notes – general form. each of the four types of equations we have studied so far this chapter;...
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9.6 Notes – General Form
Each of the four types of equations we have studied so far this chapter; circle, parabola, ellipse, and hyperbola, can be described by slicing a double-right circular cone. The most general form of any conic can be written as:
In this general form, the capital letters are just coefficients and our goal will be to transform any equation of this type into a standard form that we recognize.
2 2 0.Ax By Cx Dy E
Shape General Form: Standard Form Picture
Parabola: horizontal
Parabola: vertical
Features of graph:
Clues for Identifying Conic Types: 2 2 0.Ax By Cx Dy E
A = 0
2y a x h k
2x a y k h
Vertex: (h, k)
Only y squared
B = 0Only x squared
Additional Points: 2d from focus
1
4d
a
a > 0 open up or right, a < 0 opens down or left
Shape General Form Standard Form Picture
Circle
Features of graph:
Clues for Identifying Conic Types: 2 2 0.Ax By Cx Dy E
A = B 2 2 2x h y k r
(same denominator)
Center: (h, k) r = Radius
Shape General Form: Standard Form Picture
Ellipse: horizontal
Ellipse: vertical
Features of graph:
Clues for Identifying Conic Types: 2 2 0.Ax By Cx Dy E
A B
Center: (h, k)
Both positive
a is always the biggest denominator
2 2
2 21
x h y k
a b
2 2
2 21
x h y k
b a
A B
Both positive
Foci: c2 = a2 – b2
Shape General Form: Standard Form Picture
Hyperbola: horizontal
Hyperbola : vertical
Features of graph:
Clues for Identifying Conic Types:2 2 0.Ax By Cx Dy E
A B
Center: (h, k)
One negative
a is always first denominator, make rectangle and asymptotes
A B
One negative
Foci: c2 = a2 + b2
(x h)2
a2(y k)2
b21
(y k)2
a2(x h)2
b21
by x h k
a
ay x h k
b
***Note: The vertex or center is always at __________.
The h is with _____ and the k is with ________.
When in parentheses, take the _____________.
(h, k)
x y
opposite
2x a y k h
2y a x h k
2 2 2x h y k r
2 2
2 21
x h y k
a b
2 2
2 21
x h y k
b a
(x h)2
a2(y k)2
b21
(y k)2
a2(x h)2
b21
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
2 4 12 8 0y y x
212 4 ___ 8 ___x y y 4 4
212 2 12x y
212 2 12x y 12 12 12
212 1
12x y
1.
Parabola
1. Graph
11
412
3
3
212 1
12x y
Opens: left
Vertex: (1, 2)
d =
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
2. 2 2 2 6 6 0x y x y
2 22 ___ 6 ___ 6 ___ ___x x y y 1 19 9
2 21 3 4x y
Circle
2 21 3 4x y
Center: (1, –3)
Radius: 2
2. Graph
2 225 100 2 76 0x y x y
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
3.
2 225 100 2 76x x y y
2 225 4 ___ 2 ___ 76 ___ ___x x y y 4 1001 1
2 225 2 1 25x y
25 25 25
2 22 1
11 25
x y Ellipse
2 22 1
11 25
x y
3. Graph
a = 5b = 1
vertical
Center: (2, 1)
2 225 14 100 76 0x y x y
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
4.
2 214 25 100 76x x y y
2 214 ___ 25 4 ___ 76 ___ ___x x y y 49 494 -100
2 27 25 2 25x y
25 25 25
2 27 2
125 1
x y hyperbola
2 27 2
125 1
x y
4. Graph
horizontal
a = 5b = 1
Center: (7, 2)
2 4 8 12 0x x y
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
5.
2 4 ___ 8 12 ___x x y 4 4
22 8 8x y
-8 -8 -8 28 2 8y x
212 1
8y x
Parabola
5. Graph
11
48
2
2
Opens: up
Vertex: (-2, 1)
d =
212 1
8y x
2 24 10 40 121 0x y x y
Determine the conic and write in standard form. Complete squares as necessary. Then sketch the graph.
6.
2 210 4 40 121x x y y
2 210 ___ 4 10 ___ 121 ___ ___x x y y 25
2 25 4 5 4x y
25
2 25 5
14 1
x y Ellipse
25 100
6. Graph
a = 2b = 1
horizontal
Center: (5, 5)
2 25 5
14 1
x y