Section 9.1 Conics

OBJECTIVE 1

The Parabola .

Focus (0,p) Vertex (h,k) Geometric Definition of a Parabola: The collection of all the points P(x,y) in a a plane that are the same distance from a fixed point, the focus, as they are from a fixed line called the directrix. P

Focus (0,p) p 2p Vertex (h,k) p Directrix y = -p And the equation is… As you can plainly see the distance from the focus to the vertex is a and is the same distance from the vertex to the directrix! Neato! Focus (0,p) p 2p Vertex (h,k) p Directrix y = -p And the equation is…

Directrix p Vertex (h,k) p Focus (0,-p) And the equation is…

2p Directrix p p Focus (p,0) Vertex (h,k) And the equation is…

p p Directrix Vertex (h,k) Focus (-p,0) And the equation is…

STANDARD FORMS I like to call standard form “Good Graphing Form”

A parabola that opens up/down: A Couple More Things………… A parabola that opens up/down: Has a vertex at (h , k) Has an Axis of Symmetry at x = h 3) Coordinates of focus (h , k+p) 4) Equation of Directrix y = k-p A parabola that opens right/left: Has a vertex at (h , k) Has an Axis of Symmetry at y = k 3) Coordinates of focus (h+p , k) 4) Equation of Directrix x = h-p

General Form of any Parabola *Where either A or B is zero! * You will use the “Completing the Square” method to go from the General Form to Standard Form,

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.

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

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)

Example #2 Writing the equation of a parabola From the graph, the vertex is at the origin, (0,0), and the directrix is 2 units away from the vertex.  The parabola opens up, so the equation is in form. Since p = 2 , the equation is (Larson, Boswell, Kanold & Stiff, 2005)

#10 Write the standard form of the equation of the parabola with the given focus or directrix with the vertex at (0,0). Focus Since the focus has to be inside the parabola and lie on the axis of symmetry, this parabola opens up, and is the form The distance p is the distance from the vertex to the focus, or in this case 3.

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