Parabolas

Definitions

• Parabola – set of all points equidistant from a fixed line (directrix) and a fixed point (focus)

• Vertex – midpoint of segment from focus to directrix.

• Axis of Symmetry: line through the focus and vertex

• Vertical Parabola Form: (x – h)2 = 4p(y – k) • Horizontal Parabola form: (y – k)2 = 4p(x – h)

FOCUS FOCUS

VERTEX

VERTEX

DIRECTRIXDIRECTRIX

24x h p y k 2

4y k p x h

VERTICAL PARABOLA HORIZONTAL PARABOLAOpens either up or down. Up if p > 0, down if p < 0.

Opens either left or right. Right if p > 0, left if p < 0.

p: distance from the vertex to the focus and from the vertex to the directrix

p

p

2p 2p

Important note: The points on a parabola are symmetric across the Axis of Symmetry. So, if you know one point, you can find the other.

Focal Chord – the line segment that goes through the focus, and has endpoints on the parabola. It’s length is 4p.

Ex 1: Write the equation of the parabola in standard form.

22 4 8 8 0y x x

24 2 ____ 2 8 ____x x y

24 8 2 8x x y

2 1 44 2 2 8x x y

24 1 2 4x y

24 1 2 2x y

2 11 2

2x y

Decide whether the parabola has vertical or horizontal axis of symmetry, and tell which way the graph opens.

2. -6x2 = 3y

3. (y – 4)2 = 3x + 1

4. 2y = (x + 1)2

Vertical

Opens down

Horizontal

Opens right

Vertical

Opens up

HINT: Which variable is squared?

Given the following information, write the equation of the parabola

5. Vertex (-2, 5); p= -½; Vertical Axis of Symmetry

6. Vertex (1, -3); p = 1/8; Vertical Axis of Symmetry

22 2( 5)x y

2 11 3

2x y 2

4 ( )x h p y k

24 ( )x h p y k

Write the formula, and fill in values

Write the formula, and fill in values

Given the following information, write the equation of the parabola

7. Vertex (6, -1); p = -1/12; Horizontal Axis of Symmetry

8. Vertex (-5, -7); p = 1; Horizontal Axis of Symmetry

Write the formula, and fill in values

Write the formula, and fill in values

27 4 5y x

2 11 6

3y x

24y k p x h

24y k p x h

Vertex:

Focus:

Directrix:

Axis:

Equation:

(0,0)

(0,3)

y = -3

x = 0

What is p?

So, 4p is…?

Is this a horizontal or vertical parabola?

Write the formula, and fill in values.

3

Vertical

4 12p

212y x 212 0 0y x

212y x

24p y k x h

positive or negative?

Why?

9

Vertex:

Focus:

Directrix:

Axis:

Equation:

(0,0)

(-2, 0)

x = 2

y = 0

What is p?

Is this a horizontal or vertical parabola?

Write the formula, and fill in values.

2

Horizontal

4 8p

2 8y x 20 8 0y x

28x y

24y k p x h

positive or negative?

Why?

10

Vertex:

Focus:

Directrix:

Axis:

Equation:

(3,2)

(3,1)

y = 3

x = 3

What is p?

Is this a horizontal or vertical parabola?

Write the formula, and fill in values.

1

Vertical

4 4p

24 2 3y x

24 2 3y x

24p y k x h

positive or negative?

Why?

11

Vertex:

Focus:

Directrix:

Axis:

Equation:

(-4,2)

(0,2)

x = -8

y = 2

What is p?

Is this a horizontal or vertical parabola?

Write the formula, and fill in values.

4

Horizontal

4 16p

216 4 2x y

216 4 2x y

24p x h y k

positive or negative?

Why?

12

13. Find the Vertex, the Focus, the Directrix, and sketch the graph

212 1 3y x

What is the vertex?

What is p?

Which way does the graph open?

Where is the focus?

Where is the directrix?

Sketch the graph.

(3, -1)

3

Up

(3, 2)

y = -4

14. Find the Vertex, the Focus, the Directrix, and sketch the graph

24 3 5x y

What is the vertex?

What is p?

Which way does the graph open?

Where is the focus?

Where is the directrix?

Sketch the graph.

(3, 5)

-1

Left

(2, 5)

x = 4

15. Find the equation if…

The Vertex is (-3, 6), and the Focus is (5, 6)What do we need for the equation?

We need the vertex (GOT IT!) and p.

Draw a sketch.

How far away from the vertex is the focus?

8Positive or Negative? Therefore, p = 8 and 4p = 32So, the equation is:

232 3 6x y

16. Find the equation if…

The Vertex is (2, -1), and the Directrix is x = 5What do we need for the equation?

We need the vertex (GOT IT!) and p.

Draw a sketch.

How far away from the vertex is the directrix?

3Positive or Negative?Therefore, p = -3 and 4p = -12So, the equation is:

212 2 1x y

17. Find the equation if…

The Directrix is y = 5, and the Focus is (-3, 1)What do we need for the equation?

We need the vertex and p.

Draw a sketch. Where is the vertex in relation to the focus and directrix?

Halfway in betweenTherefore, the vertex is at

(-3, 3)Opens up or down?Therefore, p = -2 and 4p = -8So, the equation is:

28 3 3y x

Down

How does a Parabola Work?

Anything entering the parabola is reflected to the focus, concentrating

the signal.

Anything leaving from the focus reflects off the parabola in a straight line creating a

beam.

How does a Parabola Work?

18. Where are Parabolas used?

Where are Parabolas used?Cars - headlights

Sports – the trajectory of a ball

Where are Parabolas used?Communication – a microphone

Architecture – the St. Louis Arch