Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley CHAPTER 7: Conic Sections 7.1 The Parabola 7.2 The Circle and the Ellipse 7.3 The Hyperbola 7.4 Nonlinear Systems of Equations and Inequalities

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 7.1 The Parabola  Given an equation of a parabola, complete the square, if necessary, and then find the vertex, the focus, and the directrix and graph the parabola.

Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Conic Sections A conic section is formed when a right circular cone with two parts, called nappes is intersected by a plane. One of four types of curves can be formed: a parabola, a circle, an ellipse, or a hyperbola.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Parabolas A parabola is the set of all points in a plane equidistant from a fixed line (the directrix) and a fixed point not on the line (the focus). The line that is perpendicular to the directrix and contains the focus is the axis of symmetry. The vertex is the midpoint of the segment between the focus and the directrix.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Parabola with Vertex at the Origin The standard equation of a parabola with vertex (0, 0) and directrix y =  p is x 2 = 4py. The focus is (0, p) and the y-axis is the axis of symmetry.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Parabola with Vertex at the Origin The standard equation of a parabola with vertex (0, 0) and directrix x =  p is y 2 = 4px. The focus is (p, 0) and the x-axis is the axis of symmetry.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find the focus, and the directrix of the parabola Then graph the parabola. Solution: We write in the form x 2 = 4py: Thus, p =  3. Focus is (0, p), or (0,  3). Directrix is y =  p = 3.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example Find an equation of the parabola with vertex (0, 0) and focus (5, 0). Then graph the parabola. Since the focus (5, 0) is 5 units to the right of the vertex, p = 5 and the equation is y 2 = 4(5)x, or y 2 = 20x. Solution: The focus is on the x-axis so the line of symmetry is the x-axis. Thus the equation is of the type y 2 = 4px.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Parabola with Vertex (h, k) and Vertical Axis of Symmetry The standard equation of a parabola with vertex (h, k) and vertical axis of symmetry is where the vertex is (h, k), the focus is (h, k + p), and the directrix is y = k – p.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Standard Equation of a Parabola with Vertex (h, k) and Horizontal Axis of Symmetry The standard equation of a parabola with vertex (h, k) and horizontal axis of symmetry is where the vertex is (h, k), the focus is (h + p, k), and the directrix is x = h – p.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the parabola x 2 + 6x + 4y + 5 = 0, find the vertex, the focus, and the directrix. Then draw the graph. Solution: We first complete the square:

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) We see that h = –3, k = 1 and p = –1, so we have the following: Vertex (h, k): (–3, 1); Focus (h, k + p): (–3, 1 + (–1)) or (3, 0); Directrix y = k  p: y = 1  (–1), or y = 2.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example For the parabola y 2 – 2y – 8x – 31 = 0, find the vertex, the focus, and the directrix. Then draw the graph. Solution: We first complete the square:

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Example (continued) We see that h = –4, k = 1 and p = 2, so we have the following: Vertex (h, k): (–4, 1); Focus (h, k + p): (–4 + 2, 1) or (–2, 1); Directrix x = h  p: y = –4  2, or x = –6.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications Cross sections of car headlights, flashlights, search lights: the bulb is at the focus, light rays from that point are reflected outward parallel to axis of symmetry.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications (continued) Cross sections of satellite dishes, field microphones: incoming radio waves or sound waves parallel to the axis of symmetry are reflected into the focus.

Slide Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Applications (continued) Cables hung between structures in suspension bridges, such as the Golden Gate Bridge. When a cable supports only its own weight, it forms a curve called a catenary.