The Parabola 3.6 Chapter 3 Conics 3.6.1 MATHPOWERTM 12, WESTERN EDITION 3.6.1

The Parabola The parabola is the locus of all points in a plane that are the same distance from a line in the plane, the directrix, as from a fixed point in the plane, the focus. Point Focus = Point Directrix PF = PD The parabola has one axis of symmetry, which intersects the parabola at its vertex. | p | The distance from the vertex to the focus is | p |. The distance from the directrix to the vertex is also | p |. 3.6.2

The Standard Form of the Equation of a Parabola with Vertex (0, 0) vertex (0, 0) and focus on the x-axis is y2 = 4px. The coordinates of the focus are (p, 0). The equation of the directrix is x = -p. If p > 0, the parabola opens right. If p < 0, the parabola opens left. 3.6.3

The Standard Form of the Equation of a Parabola with Vertex (0, 0) vertex (0, 0) and focus on the y-axis is x2 = 4py. The coordinates of the focus are (0, p). The equation of the directrix is y = -p. If p > 0, the parabola opens up. If p < 0, the parabola opens down. 3.6.4

A parabola has the equation y2 = -8x. Sketch the Sketching a Parabola A parabola has the equation y2 = -8x. Sketch the parabola showing the coordinates of the focus and the equation of the directrix. The vertex of the parabola is (0, 0). The focus is on the x-axis. Therefore, the standard equation is y2 = 4px. Hence, 4p = -8 p = -2. The coordinates of the focus are (-2, 0). F(-2, 0) The equation of the directrix is x = -p, therefore, x = 2. x = 2 3.6.5

Finding the Equation of a Parabola with Vertex (0, 0) A parabola has vertex (0, 0) and the focus on an axis. Write the equation of each parabola. a) The focus is (-6, 0). Since the focus is (-6, 0), the equation of the parabola is y2 = 4px. p is equal to the distance from the vertex to the focus, therefore p = -6. The equation of the parabola is y2 = -24x. b) The directrix is defined by x = 5. Since the focus is on the x-axis, the equation of the parabola is y2 = 4px. The equation of the directrix is x = -p, therefore -p = 5 or p = -5. The equation of the parabola is y2 = -20x. c) The focus is (0, 3). Since the focus is (0, 3), the equation of the parabola is x2 = 4py. p is equal to the distance from the vertex to the focus, therefore p = 3. The equation of the parabola is x2 = 12y. 3.6.6

The Standard Form of the Equation with Vertex (h, k) For a parabola with the axis of symmetry parallel to the y-axis and vertex at (h, k): The equation of the axis of symmetry is x = h. The coordinates of the focus are (h, k + p). The equation of the directrix is y = k - p. When p is positive, the parabola opens upward. When p is negative, the parabola opens downward. The standard form for parabolas parallel to the y-axis is: (x - h)2 = 4p(y - k) The general form of the parabola is Ax2 + Cy2 + Dx + Ey + F = 0 where A = 0 or C = 0. 3.6.7

The Standard Form of the Equation with Vertex (h, k) For a parabola with an axis of symmetry parallel to the x-axis and a vertex at (h, k): The equation of the axis of symmetry is y = k. The coordinates of the focus are (h + p, k). The equation of the directrix is x = h - p. When p is positive, the parabola opens to the right. When p is negative, the parabola opens to the left. The standard form for parabolas parallel to the x-axis is: (y - k)2 = 4p(x - h) 3.6.8

Finding the Equations of Parabolas Write the equation of the parabola with a focus at (3, 5) and the directrix at x = 9, in standard form and general form The distance from the focus to the directrix is 6 units, therefore, 2p = -6, p = -3. Thus, the vertex is (6, 5). The axis of symmetry is parallel to the x-axis: (y - k)2 = 4p(x - h) h = 6 and k = 5 (y - 5)2 = 4(-3)(x - 6) (y - 5)2 = -12(x - 6) (6, 5) Standard form y2 - 10y + 25 = -12x + 72 y2 + 12x - 10y - 47 = 0 General form 3.6.9

Finding the Equations of Parabolas Find the equation of the parabola that has a minimum at (-2, 6) and passes through the point (2, 8). The axis of symmetry is parallel to the y-axis. The vertex is (-2, 6), therefore, h = -2 and k = 6. Substitute into the standard form of the equation and solve for p: (x - h)2 = 4p(y - k) x = 2 and y = 8 (2 - (-2))2 = 4p(8 - 6) 16 = 8p 2 = p (x - h)2 = 4p(y - k) (x - (-2))2 = 4(2)(y - 6) (x + 2)2 = 8(y - 6) Standard form x2 + 4x + 4 = 8y - 48 x2 + 4x + 8y + 52 = 0 General form 3.6.10

Find the coordinates of the vertex and focus, Analyzing a Parabola Find the coordinates of the vertex and focus, the equation of the directrix, the axis of symmetry, and the direction of opening of y2 - 8x - 2y - 15 = 0. 4p = 8 p = 2 y2 - 8x - 2y - 15 = 0 y2 - 2y + _____ = 8x + 15 + _____ 1 1 (y - 1)2 = 8x + 16 (y - 1)2 = 8(x + 2) Standard form The vertex is (-2, 1). The focus is (0, 1). The equation of the directrix is x + 4 = 0. The axis of symmetry is y - 1 = 0. The parabola opens to the right. 3.6.11

Graphing a Parabola y2 - 10x + 6y - 11 = 0 y2 + 6y + _____ = 10x + 11 + _____ 9 9 (y + 3)2 = 10x + 20 (y + 3)2 = 10(x + 2) 3.6.12

General Effects of the Parameters A and C When A x C = 0, the resulting conic is an parabola. When A is zero: If C is positive, the parabola opens to the left. If C is negative, the parabola opens to the right. When C is zero: If A is positive, the parabola opens up. If A is negative, the parabola opens down. When A = D = 0, or when C = E = 0, a degenerate occurs. E.g., x2 + 5x + 6 = 0 x2 + 5x + 6 = 0 (x + 3)(x + 2) = 0 x + 3 = 0 or x + 2 = 0 x = -3 x = -2 The result is two vertical, parallel lines. 3.6.13

Assignment 3.6.14