Parabolas

Advanced GeometryConic Sections

Lesson 3

Definition – the set of all points in a plane that are the same distance from a given point, called the focus, and a given line called the directrix

p

Axis of Symmetry

DirectrixV

F

p

Horizontal Axis Vertical Axis of Symmetry of Symmetry

Equation of a Parabola

Vertex

Focus

Axis of Symmetry

Directrix

Direction of Opening

2y a x h k 2x a y k h

( , )h k ( , )h k

( , )h p k( , )h k p

y kx hx h p y k p

0

0

right if p

left if p

0

0

up if p

down if p

1

4p

a

Example:For the equation of each parabola, find the coordinates of the vertex and focus, the equations of the directrix and axis of symmetry. Then graph the equation.

212 1

16y x

Example:For the equation of each parabola, find the coordinates of the vertex and focus, the equations of the directrix and axis of symmetry. Then graph the equation.

216

8x y

Example:Using the graph below, write the equation for the parabola.

Example:Using the graph below, write the equation for the parabola.

Example:Write the equation of the parabola that meets each set of conditions.

The vertex is (-4, 5), the parabola opens to the left, and the focus is 5 units from the vertex.

Example:Write the equation of the parabola that meets each set of conditions.

The parabola has a focus of

and a minimum point at (-1, -1).

11,2

Example:Solve for x.

21 2 8y x