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