9.2 Notes – Parabolas

Vertex:

Axis of Symmetry:

Focus:

Directrix:

Turning point

Through vertex, folds in half

Point inside

Line outside, perpendicular to axis of symmetry

Parts of a Parabola: (U-shaped)

vertex

Axis of symmetry

focus

directrix

Parabola: equidistant to the Focus and Directrix

Remember transformations of parabolas?

Vertex: ____________ (h, k) 2y a x h k

To find the distance from the vertex to the directrix or focus:

1

4d

a

1

4a

d or

Make sure to take the absolute value for distance!

2y a x h k

Using y = ax2 where “a” determines the width and direction of the graph:

If:

width is: ________ _________ ________

1 1 1a a a

normal skinny wide

Where k is the ___________ shift of vertex from (0, 0) and h is the _____________ shift

Vertex: _____________

a > 0 parabola opens _______ a > 0 parabola opens to the ______

a < 0 parabola opens _______ a < 0 parabola opens to the ______

Axis of symmetry: _________ Axis of symmetry: _____________

2y a x h k 2

x a y k h

verticalhorizontal

(h, k)

up

down

y = k

right

left

x = h

1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens:

a) 21( 7) 2

9x y 2

x a y k h

Vertex: _____________

Horizontal or Vertical?

Opens: _____________

(–2, –7)

left

1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens:

b)

Vertex: _____________

Horizontal or Vertical?

Opens: _____________

(2, –3)

up

22( 2) 3y x 2y a x h k

1. Identify vertex of each parabola, whether it’s vertical or horizontal, and which way it opens:

c) 2x a y k h

Vertex: _____________

Horizontal or Vertical?

Opens: _____________

(2, –5)

right

23 ( 5) 6x y 23 ( 5) 6x y

21( 5) 2

3x y

2. Draw the directrix |d| behind the parabola. Show where the focus is by going along the axis of symmetry d units toward the inside of the parabola.

21

8y x

x y

-8

-4

-2

0

2

4

8

1

4d

a = 1

14

8

2

112

11

2 1 2 2 Focus: (0, 2)

y = -2

1/20

2

8

1/2

2

8

Try any point on the parabola and see if the distance to the focus is about the same distance to the directrix you drew.

Notice two special points on any parabola can be drawn once you know the vertex and the distance d from the vertex to the directrix.

Additional points: 2d

2. Graph the parabola.

Vertical or horizontal

Opens ___________

Vertex ( , )

Axis of symmetry: _______

d =

21( 5) 2

4y x

down5 2

x = 511

44

Focus ( , )Directrix: __________Additional points ( , ) ( , )

1

5 1y = 3

3 1 7 1

3. Graph the parabola.

Vertical or horizontal

Opens ___________

Vertex ( , )

Axis of symmetry: _______

d =

left4 2

y = 211

412

Focus ( , )Directrix: __________Additional points ( , ) ( , )

3

1 2x = 7

1 –4 1 8

21( 2) 4

12x y

3

4. Graph the parabola.

Vertical or horizontal

Opens ___________

Vertex ( , )

Axis of symmetry: _______

d =

right–2 –1

y = –111

48

Focus ( , )Directrix: __________Additional points ( , ) ( , )

2

0 –1x = –4

1 –4 1 8

2

28 16 ( 1)x y 28 ( 1) 16x y

21( 1) 2

8x y

5. Graph the parabola.

2 10 6 7 0x x y

2 10 ____ 6 7 _____x x y

2 10 6 7x x y

25 25

25 6 18x y

26 5 18y x

215 3

6y x

5. Graph the parabola.

Vertical or horizontal

Opens ___________

Vertex ( , )

Axis of symmetry: _______

d =

up5 –6

x = 511

46

Focus ( , )Directrix: __________Additional points ( , ) ( , )

5 -4.5y = -7.5

2 -4.5 8 -4.5

123

3

23

2

215 3

6y x

6. Graph the parabola.

28 2 ____ 15 _____x y y

28 2 15x y y

1 1

28 1 16x y

211 2

8x y

2 8 2 15 0y x y

28 1 16x y

3. Graph the parabola.

Vertical or horizontal

Opens ___________

Vertex ( , )

Axis of symmetry: _______

d =

left2 1

y = 111

48

Focus ( , )Directrix: __________Additional points ( , ) ( , )

2

0 1x = 4

1 –4 1 8

2

211 2

8x y

7. Write the standard equation for the parabola that has a vertex of (-1, -5) and a focus of (-1, -6). First, show a rough sketch of the layout only and determine whether a will be positive or negative, then find a.

2y a x h k

21 5y a x

1

4a

d

1

4 1

1

4

211 5

4y x

8. Write the standard equation for the parabola that has a vertex of (3, 1) and a focus of (6, 1). First, show a rough sketch of the layout only and determine whether a will be positive or negative, then find a.

1

4a

d

1

4 3

1

12

2x a y k h

21 3x a y

211 3

12x y

Ax2 + Cy2 + Dx + Ey + F = 0

How is the parabola equation different from circles?

Circle Parabola

x and y are squared

Either x or y are squared, not both

Has an A and C term A or C = 0