1. MTH065 Elementary Algebra II Chapter 13 Conic Sections Introduction Parabolas (13.1) Circles (13.1) Ellipses (13.2) Hyperbolas (13.3) Summary

2. Where we’ve been … MTH 060 – Linear Functions & Equations Single Variable: ax + b = 0 Solution: A single real number. Two Variables: ax + by = c y = mx + b f(x) = mx + b Solutions: Many ordered pairs of real numbers. Graph: A line.

3. Where we’ve been … MTH 065 – Quadratic Functions & Equations Single Variable: ax 2 + bx + c = 0 Solutions: 0, 1, or 2 real numbers Two Variables: y = ax 2 + bx + c f(x) = ax 2 + bx + c f(x) = a(x – h) 2 + k Solutions : Many ordered pairs of real numbers. Graph: A parabola.

4. What’s missing … Quadratic Equations that may also include a y 2 term (not all functions). Ax 2 + By 2 + Cx + Dy + E = 0 A, B, C, D, & E are constants A and B not both 0 Note: Quadratic equations may also include an xy term, but the study of such equations requires trigonometry.

5. Parabolas y = ax 2 + bx + c Graphing (complete the square): y = a(x - h) 2 + k Vertex: (h, k) h = -b/(2a) Orientation: Open upward: a > 0 Open downward: a < 0 Width: Narrow: |a| > 1 Wide: |a| < 1 Graphing: Vertex & One Other Point

6. Parabolas x = ay 2 + by + c Graphing (complete the square): x = a(y - k) 2 + h Vertex: (h, k) k = -b/(2a) Orientation: Open right: a > 0 Open left: a < 0 Width: Narrow: |a| > 1 Wide: |a| < 1 Graphing: Vertex & One Other Point

7. Parabolas – Special Properties Focus The point 1/(4a) units from the vertex along the axis of symmetry and inside the parabola. Reflective property: Light or any other wave emitted from the focus will be reflected in a beam parallel to the axis of symmetry. A satellite dish, for example, uses this property in reverse.

8. Ellipses Ax 2 + By 2 + Cx + Dy + E = 0 where A & B are both positive or both negative. Graphing form: Complete the squares & set equal to 1 Center: (h,k) 4 Vertices: (h ± a, k), (h, k ± b)

9. Ellipses – Special Properties Foci The two points c units from the center along the major axis where c 2 = a 2 – b 2 if a > b or c 2 = b 2 – a 2 if a < b. Reflective property: Sound or any other wave emitted from one focus will be reflected to the other focus. Satellites have elliptical orbits with the object being orbited at one of the foci.

10. Circles – Special Ellipses A circle is just an ellipse with a = b and a single “focus” at the center (since c 2 = a 2 – b 2 = 0). Ax 2 + Ay 2 + Cx + Dy + E = 0 (x – h) 2 + (y – k) 2 = r 2 Center: (h, k) Radius: r

11. Hyperbolas Ax 2 + By 2 + Cx + Dy + E = 0 where A & B have opposite signs. Graphing form: Complete the squares & set equal to 1 Center: (h,k) 2 Vertices: 1 st form: (h ± a, k) 2 nd form: (h, k ± b) Asymptotes: or b a (h,k) b a

12. Hyperbolas – Special Properties Foci The two points c units from the center inside each branch, where c 2 = a 2 + b 2 Reflective property: Light or any other wave emitted from one focus towards the other branch will be reflected directly away from the other focus (or vice versa). Hyperbolic mirrors are used in reflector telescopes. Lampshades cast hyperbolic shadows on a wall. Parabola Hyperbola

13. Conic Sections – Summary Ax 2 + By 2 + Cx + Dy + E = 0 A ≠ 0 & B = 0 Up/Down Parabola A = 0 & B ≠ 0 Left/Right Parabola A & B w/ same sign Ellipse A = B gives a circle A & B w/ opposite signs Hyperbola To graph … complete the squares.

14. More Applications of Conics Parabolas http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/appparab.html http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/appparab.html Ellipses http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/appellip.html http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/appellip.html Hyperbolas http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/apphyper.html http://www.doe.virginia.gov/Div/Winchester/jhhs /math/lessons/calc2004/apphyper.html