Section 10.1 Parabolas

Objectives:•To define parabolas geometrically.•Use the equation of parabolas to find relevant information.•To find the equation of parabolas given certain information

Parabola—Geometric Definition• A parabola is the set of points in the plane

equidistant from a fixed point F (focus) and a fixed line l (directrix).

– The vertex V lies halfway between the focus and the directrix.

– The axis of symmetry is the line that runs through the focus perpendicular to the directrix.

Parabola with Vertical Axis of SymmetryThe graph of the equation y = ax2 is a

parabola with these properties. • vertex: V(0,0)• focus: F(0, p) where p is the distance

between the focus and vertex• directrix: y = -p • a = 1

4p(recall: a is the number that determines how wide or narrow the parabola is)

Parabola with Vertical Axis• The parabola opens:

– Upward if p > 0.– Downward if p < 0.

Ex 1. Find the equation of the parabola with vertex V(0,0) and focus F(0,2).

Ex 2. Find the equation of the parabola with vertex (0,0) and directrix y = 4.

Class Work 1. Find the equation of the parabola with focus

(0,-5) and vertex (0,0).

2. Find the equation of the parabola with focus (0,3) and directrix y = -3

The equation of the parabola whose vertex is (h,k) is

Parabolas whose vertex is not at the origin:

2( )y a x h k

Ex 3. Find the equation of the parabola with focus (3, -1) and vertex (3, -4).

Ex. 4 Find the equation of the parabola with focus (4, -1) and vertex (4, 1).

Class Work

3. Find the equation of the parabola with vertex (2, 8) and focus (2, 3).

4. Find the equation of the parabola with focus (-1, -3) and vertex (-1, 1)

Parabola with Horizontal AxisThe graph of the equation x=ay2

is a parabola with these properties:

• Vertex: V(0,0)

• Focus: F(p, 0)

• directrix: x = -p

Parabola with Horizontal Axis• The parabola opens:

– To the right if p > 0.– To the left if p < 0.

Ex 5. Find the equation of the parabola with focus (6, 0) and vertex (0, 0).

Class Work

5. Find the equation of the parabola with focus (-3,0) and directrix x = 3

Parabolas whose vertex is not at the origin:

The equation of the parabola whose vertex is (h,k) is

2( )x a y k h

Ex 6. Find the equation of the parabola with vertex (4, -2) and focus (2, -2)

Class Work6. Find the equation of a parabola with focus

(3,2) and vertex (5, 2).

7. Find the equation of a parabola with vertex (-4,1) and directrix x = -7.

HW#1 Parabolas Worksheet