10.2The Parabola

A parabola is defined as the collection of all points P in the plane that are the same distance from a fixed point F as they are from a fixed line D. The point F is called the focus of the parabola, and the line D is its directrix. As a result, a parabola is the set of points P for which

d(F, P) = d(P, D)

D: x = -a

F = (a, 0)x

y

V

D: x = a

F: (-a, 0)

V

y

x

y

x

D: y = -aV

F: (0, a)

x

y

D: y = a

F: (0, -a)

Find an equation of the parabola with vertex at the origin and focus (-2, 0). Graph the equation by hand and using a graphing utility.

Vertex: (0, 0); Focus: (-2, 0) = (-a, 0)

V=(0,0)F=(-2,0)

The line segment joining the two points above and below the focus is called the latus rectum.

Let x = -2 (the x-coordinate of the focus)

The points defining the latus rectum are (-2, -4) and (-2, 4).

Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Right, a > 0.

F = (h + a, k)

V = (h, k)

D: x = -a + hy

x

Axis of symmetry

y = k

Parabola with Axis of Symmetry Parallel to x-Axis, Opens to the Left, a > 0.

D: x = a + h

F = (h - a, k)

Axis of symmetry y = k

y

x

V = (h, k)

Parabola with Axis of Symmetry Parallel to y-Axis, Opens up, a > 0.

D: y = - a + k

F = (h, k + a)

V = (h, k)

y

x

Axis of symmetry x = h

Parabola with Axis of Symmetry Parallel to y-Axis, Opens down, a > 0.

y

x

D: y = a + k

F = (h, k - a)

V = (h, k)

Axis of symmetry x = h

Complete thesquare

Vertex: (h, k) = (-2, -3)

a = 2

Focus: (-2, -3 + 2) = (-2, -1)

Directrix: y = -a + k = -2 + -3 = -5

Latus Rectum: Let y = -1

(-6, -1) or (2, -1)

10 0 10

10

10

(-2, -3)(-2, -1)

y = -5

(-6, -1)

(2, -1)