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