1 / 12

The Parabola

Unit 5 Conics. The Parabola. 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

aine
Download Presentation

The Parabola

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Unit 5 Conics.. The Parabola

  2. 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. The Standard Form of the Equation of a Parabola with Vertex (0, 0) The equation of a parabola with vertex (0, 0) • Standard Equationx2 = 4py y2= 4px • Opensup 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

  4. 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

  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 isy2 = -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 = 5or p = -5. The equation of the parabola isy2 = -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 isx2 = 12y.

  6. The Standard Form of the Equation with Vertex (h, k) • standard form • axis of symmetry x= h y = k • opens up or down right or left • focus (h, k + p) (h + p, k) • directrixy = k – p x = h - p (x - h)2 = 4p(y - k) (y - k)2 = 4p(x - h) The general form of the parabola is Ax2 + Cy2 + Dx + Ey + F = 0 where A = 0 or C = 0.

  7. 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

  8. 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

  9. 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.

  10. 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)

  11. General Effects of the Parameters A and C When AxC = 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.

  12. Assignment Suggested Questions: Page 639 1, 3, 4, 7-10, 12-34 even, 50, 52.

More Related