The Conic Sections Section 3.1 The Parabola
Definition: Parabola A parabola is the set of all points in the plane that are equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus).
The Parabola Axis of symmetry can be described as the line perpendicular to the directrix and containing the focus. Vertex is the point on the axis of symmetry that is equidistant from the focus and directrix.
The Parabola If we use the coordinates (h, k) for the vertex, then the focus is (h, k + p) and the directrix is y = k p, where p (the focal length) is the directed distance from the vertex to the focus. If the focus is above the vertex, then p > 0, and if the focus is below the vertex, then p < 0. The distance from the vertex to the focus or from the vertex to the directrix is p.
The Parabola The distance d 1 from an arbitrary point (x, y) on the parabola to the directrix is the distance from (x, y) to (x, k p). We use the distance formula to find d 1 : Now we find the distance d 2 between (x, y) and the focus (h, k + p): Since d 1 = d 2 for every point (x, y) on the parabola, we have the following equation. Verify that squaring each side and simplifying yields
Theorem: The Equation of a Parabola The equation of a parabola with focus (h, k + p) and directrix y = k p is y = a(x h) 2 + k, where a = 1/(4p) and (h, k) is the vertex.
The Parabola The link between the geometric definition and the equation of a parabola is a = 1/(4p). For any particular parabola, a and p have the same sign. If they are both positive, the parabola opens upward and the focus is above the directrix. If they are both negative, the parabola opens downward and the focus is below the directrix. Since a is inversely proportional to p, smaller values of p correspond to larger values of a and to narrower parabolas.
Graphing Parabola: Vertex Form Graph f x = 2(x 3) 2 + 1 1. Identify Vertex 2. Create Table of Values x F(x) Vertex
Graphing Parabola: Vertex Form f x = 4(x + 4) 2 3 f x = 3(x 3) 2 4 x F(x) x F(x)
Graphing Parabola: Standard Form Graph f x = x 2 4x 1 1. Change into standard form by completing the square 2. Create Table of Values x F(x) Vertex
Graphing Parabola: Standard Form Graph f x = x 2 4x 1 x F(x) Vertex
Change from Standard to Vertex Form f x = 2x 2 20x + 3
Standard Equation of Parabola (using vertex and focal length) The equation of a (vertical) parabola with vertex (h, k) and focal length p is (x h) 2 = 4p(y k) If p > 0, the parabola opens upwards; if p < 0, it opens downwards 13 Presentation Title runs here l 00/00/00
Example 1: Find vertex, focus and directrix (x 1) 2 = 4(y + 3) 1. Find h, k, p (x h) 2 = 4p(y k) Vertex Focus Directrix 14 Presentation Title runs here l 00/00/00
Standard Equation of Parabola (using vertex and focal length) The equation of a (horizontal) parabola with vertex (h, k) and focal length p is (y k) 2 = 4p(x h) If p > 0, the parabola opens right; if p < 0, it opens left 15 Presentation Title runs here l 00/00/00
Write Equation Using Vertex and Focus Write equation of the parabola with vertex (7, 8) and focus (7, 3). Then write equation of directrix. Solution: 16 Presentation Title runs here l 00/00/00
Parabolas Opening to the Left or Right The graphs of y = 2x 2 and x = 2y 2 are both parabolas. Interchanging the variables simply changes the roles of the x- and y-axes. The parabola y = 2x 2 opens upward, whereas the parabola x = 2y 2 opens to the right. For parabolas opening right or left, the directrix is a vertical line. If the focus is to the right of the directrix, then the parabola opens to the right. If the focus is to the left of the directrix, then the parabola opens to the left.
Standard Equations of Parabolas Vertical Horizontal Equation (x h) 2 = 4p(y k) x 2 = 4py (y k) 2 = 4p(x h) y 2 = 4px Vertex (h, k) or (0, 0) h, k or (0, 0) Focus (h, k + p) (h + p, k) Directrix y = k p x = h p Axis of Symmetry Focal Diameter x = h 4p y = h 4p Opens upward if p > 0; downward if p < 0 right if p > 0; left if p < 0