Unit 5 Conics..

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 |. The Parabola

The Standard Form of the Equation of a Parabola with Vertex (0, 0) Standard Equation x 2 = 4py y 2 = 4px Opens up or down Right or left Focus (0,p) (p,0) Directrix y = -p x = -p Axis y – axis x – axis Focal Length p p Focal Width The equation of a parabola with vertex (0, 0)

A parabola has the equation y 2 = -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 y 2 = 4px. Hence, 4p = -8 p = -2. The coordinates of the focus are (-2, 0). The equation of the directrix is x = -p, therefore, x = 2. F(-2, 0) x = 2 Sketching a Parabola

A parabola has vertex (0, 0) and the focus on an axis. Write the equation of each parabola. Since the focus is (-6, 0), the equation of the parabola is y 2 = 4px. p is equal to the distance from the vertex to the focus, therefore p = -6. The equation of the parabola is y 2 = -24x. b) The directrix is defined by x = 5. The equation of the directrix is x = -p, therefore -p = 5 or p = -5. The equation of the parabola is y 2 = -20x. Finding the Equation of a Parabola with Vertex (0, 0) Since the focus is on the x-axis, the equation of the parabola is y 2 = 4px. c) The focus is (0, 3). a) The focus is (-6, 0). Since the focus is (0, 3), the equation of the parabola is x 2 = 4py. p is equal to the distance from the vertex to the focus, therefore p = 3. The equation of the parabola is x 2 = 12y.

standard form axis of symmetry x = h y = k opens up or downright or left focus (h, k + p) (h + p, k) directrix y = k – p x = h - p (x - h) 2 = 4p(y - k) The general form of the parabola is Ax 2 + Cy 2 + Dx + Ey + F = 0 where A = 0 or C = 0. The Standard Form of the Equation with Vertex (h, k) (y - k) 2 = 4p(x - h)

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). (6, 5) The axis of symmetry is parallel to the x-axis: (y - k) 2 = 4p(x - h)h = 6 and k = 5 Standard form y y + 25 = -12x + 72 y x - 10y - 47 = 0 General form (y - 5) 2 = 4(-3)(x - 6) (y - 5) 2 = -12(x - 6)

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) (2 - (-2)) 2 = 4p(8 - 6) 16 = 8p 2 = p x = 2 and y = 8 (x - h) 2 = 4p(y - k) (x - (-2)) 2 = 4(2)(y - 6) (x + 2) 2 = 8(y - 6) Standard form x 2 + 4x + 4 = 8y - 48 x 2 + 4x + 8y + 52 = 0 General form Finding the Equations of Parabolas

Find the coordinates of the vertex and focus, the equation of the directrix, the axis of symmetry, and the direction of opening of y 2 - 8x - 2y - 15 = 0. y 2 - 8x - 2y - 15 = 0 y 2 - 2y + _____ = 8x _____ 11 (y - 1) 2 = 8x + 16 (y - 1) 2 = 8(x + 2) 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. 4p = 8 p = 2 Standard form Analyzing a Parabola

Graphing a Parabola y x + 6y - 11 = 0 99y 2 + 6y + _____ = 10x _____ (y + 3) 2 = 10x + 20 (y + 3) 2 = 10(x + 2)

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.

Suggested Questions: Page 639 1, 3, 4, 7-10, even, 50, 52.