Analytic Geometry

1. Analytic Geometry Parabolas

2. What You Should Learn Write equations of parabolas in standard form and graph parabolas. Use the reflective property of parabolas to solve real-life problems.

3. Plan for the day… • Quick Review • Lines • Recognizing different conic sections • Circles • Parabolas (functions and non-functions) • Horizontal and vertical • Focus • Homework • Quiz next class – Lines

4. Lines

5. Lines

6. Conics

7. Conics A conic section (or simply conic) is the intersection of a in the formation of the four basic conics; the intersecting plane does not pass through the vertex of the cone. Circle Ellipse Hyperbola Parabola Basic Conics

8. Conics When the plane does pass through the vertex, the resulting figure is a degenerate conic. Line Point Two Intersecting Lines Degenerate Conics

9. Conics There are several ways to approach the study of conics. You could begin by defining conics in terms of the intersections of planes and cones, as the Greeks did, or you could define them algebraically, in terms of the general second-degree equation Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. However, you will study a third approach, in which each of the conics is defined as a locus (collection) of points satisfying a geometric property. – Example: A circle is the locus of coplanar points that are equidistant from a given point (the center).

10. Recognizing a Conic AC = 0 - A or C is zero - no x2 or no y2 term Parabola A = C - A is equal to C, same value Circle AC > 0 - A and C have the same sign but have different values Ellipse AC < 0 - A and C have different signs Hyperbola

11. Circles

12. Circles Given an standard form of a conic section, If A = C, then the equation is a circle. Standard form of a circle with the center at (0, 0) is: x2 + y2 = r2 where r is the radius. Standard form of a circle with the center at (h, k) and radius of r is:

13. Changing format for a circle to identify key information General form of a circle (A=C): Standard form of a circle: Move from general to standard form by completing the square.

14. Parabolas

15. Parabolas We have learned that the graph of the quadratic function f (x) = ax2 + bx + cis a parabola that opens upward or downward. The following definition of a parabola is more general in the sense that it is independent of the orientation of the parabola.

16. Parabolas • Orientation • Vertex (h, k) • Directrix • Focus • Axis of symmetry • “p” • Equations:

17. axis focus directrix vertex Definition of Parabola A parabola is the set of all points in the plane equidistant from a fixed line and a fixed point not on the line. The fixed line is the directrix. The fixed point is the focus. parabola The axis is the line passing through the focus and perpendicular to the directrix. The vertex is the midpoint of the line segment along the axis joining the directrix to the focus.

18. The standard form for the equation of a parabola with vertex at the origin and a vertical axis is: x2 = 4py where p 0 vertical axis:x = 0 directrix:y = –p, focus: (0, p) y x2 = 4py (0, p) p x y = –p (0, 0) Note: p is the directed distance from the vertex to the focus.

19. (0, 0) y2 = 4px y p (p, 0) x x = –p The standard form for the equation of a parabola with vertex at the origin and a horizontal axis is: Parabola with Horizontal Axis y2 = 4px where p 0 horizontal axis: y = 0, directrix: x = –p focus: ( p, 0) Note: p is the directed distance from the vertex to the focus.

20. Parabolas Using the definition of a parabola, you can derive the following standard form of the equation of a parabolawhose directrix is parallel to the x-axis or to the y-axis.

21. y y = 2 (0, 0) x (0, –2) x = 0 Example: Graph Parabola Example: Find the directrix, focus, and vertex, and sketch the parabola with equation . Rewrite the equation in standard form x2 = 4py. x2 = –8yx2 = 4(–2)y p = –2 vertex: (0, 0) vertical axis:x = 0 directrix:y = – py = 2 focus: = (0, p)  (0, –2)

22. y p = 1 x (1, 0) (0, 0) vertex x = -1 Example: Write the Equation of a Parabola Example: Write the standard form of the equation of the parabola with focus (1, 0) and directrix x = –1. Use the standard from for the equation of a parabola with a horizontal axis: y2 = 4px. p = 1 y2 = 4(1)x. The equation is y2 = 4x.

23. y (-1, 1) y = 3/2 (-1, ½ ) x x = - 1 Example: Graph Parabola Example: Find the directrix, focus, and vertex, and sketch the parabola with equation Rewrite the equation in standard form (x - h)2 = 4p(y - k). -2y = x2 + 2x – 1  1 – 2y = x2 + 2x  1 + 1 – 2y = (x2 + 2x + 1)  2 – 2y = (x + 1)2 -2(y – 1) = (x + 1)2, (h, k) = (-1, 1), p = - ½ vertex: (-1, 1) vertical axis:x = -1 directrix:y = k – py = 3/2 focus: = (h, k + p)  (-1, ½)

24. Application

25. Application Parabolas occur in a wide variety of applications. For instance, a parabolic reflector can be formed by revolving a parabola around its axis. The resulting surface has the property that all incoming rays parallel to the axis are reflected through the focus of the parabola.

26. Application This is the principle behind the construction of the parabolic mirrors used in reflecting telescopes. Conversely, the light rays emanating from the focus of aparabolic reflector used in aFlashlight are all parallel to one another.

27. Parabolas Vertical orientation Vertex (h, k) Focus (h, k + p) Directrix y = k – p Horizontal orientation Vertex (h, k) Focus (h + p, k) Directrix x = h – p

28. Skills • Recognizing a parabola • Given an equation • Find the orientation • Find the focus • Find the vertex • Find the directrix • Sketch • Given key information, find the equation in standard form

29. Examples • Find the vertex, focus and directrix • x2 – 2x + 8y + 9 = 0 • Find the standard form of the equation • Vertex: ( -1, 2), Focus (-1, 0) • Vertex: (-2, 1), Directrix: x = 1

30. Applications • Circles • Earth quake tracking • Parabolas • cable of a suspension bridge • Satellites • flashlights • parabolic reflector • path of a projectile • solar furnace

31. Homework 40 Homework • Page 712, section 10.2; #5-10 all (matching)15, 17, 21, 23, 31, 43, 47, 49, 51 Quiz next class - Lines