10.2 The 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 = -a V 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 + h y 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 the square

Vertex: (h, k) = (-2, -3) a = 2 Focus: (-2, ) = (-2, -1) Directrix: y = -a + k = = -5

Latus Rectum: Let y = -1 (-6, -1) or (2, -1)

(-2, -3) (-2, -1) y = -5 (-6, -1) (2, -1)