Understanding Conics in Coordinate System: Equations, Characteristics, and Graphs

2. Concept Category #14Conics in the Rectangular Coordinate System 6A I can derive the equations of circles, parabolas, ellipses and hyperbolas given the general equation in the second degree. I can find the key characteristics for each conic (i.e., focus, directrix, vertices, co-vertices, foci, asymptotes, minor axis, major axes). Foundational Skill: I can recognize and graph conic functions, including ellipses and hyperbolas.

3. Brainteaser: A radio tower services a 10 mile radius. You stop your car 5 miles east and 9 miles north of the tower. Will you be able to receive radio waves from the tower?

4. Task Find the equation of the circle centered at the origin (in terms of x and y) that passes through the point (7,24).

5. What do you notice?

6. Note and Notice

7. What do you notice?

8. I. Classifying Conics & Analyzing GraphsA. Definition Conic Sections: The curves formed when a plane intersects the surface of a right cylindrical cone.

9. B. Visual

10. Standard form for the equation of a circle: center: (h,k) radius: r

11. Example What is the equation for the circle shown below?

12. Graph the following function.

13. Routine • Graph the following functions: • 1. • 2. • 3. Write the center-radius equation of a circle that is • tangent to the x-axis, with a center located at (4, -6). Goal Problems: Determine choices for practice Recall & Reproduction Graph the following functions: 1. 2. 3. State the equation of a circle in general form which has a center at (5, -3) and a radius of 9.

14. Ellipses Standard Equation: OR

15. For Both Types: • Center (h,k) • a is the length of the semi-major axis (2a is the length of the major axis) • b is the length of the semi-minor axis (2b is the length of the minor axis) • c : vertices: (h, k + a) and (h, k – a) foci: (h, k + c) and (h, k – c) vertices: (h + a, k) and (h – a, k) foci: (h + c, k) and (h – c, k)

16. Hyperbolas: For Both: center: (h, k) OR a a 0 vertices: (h , k – a) and (h, k + a) foci: (h, k – c) and (h, k + c) asymptotes: vertices: (h – a, k) and (h + a, k) foci: (h – c, k) and (h + c, k) asymptotes:

17. Example: Graph the following function:

18. Example: What conic function do you think this is? Graph the following function:

19. Example: Graph the following function: 9x2-72x-16y2-32y-16=0

20. Determine the type of conic and justify your reasoning. Then sketch the graph

21. A. Definition Conics can be classified by computing the discriminant B2– 4AC, of equations of the form Ax2 + Bxy + Cy2 + Dx+ Ey + F = 0 B = 0 for the conics we will be working with

22. Example Identify each of the following conics:

23. Example Identify each of the following conics:

24. A. Definition • Parabola is the set of all points in a plane whose distance from a fixed point is equal to its distance from a fixed line. Directrix Focus

25. B. Visual

26. Recall Parabolas? Standard form for the equation of a parabola: Vertex: (h, k) Axis of symmetry: x = h focus: (h, k + p) directrix: y = – p+k where p = 1/4a Vertex: (h, k) Axis of symmetry: y = k focus: (h + p, k) directrix: x = – p+h where p = 1/4a

27. C. Process Sketch the graph of y2 + 2y + 8x + 17 = 0. Specify its vertex, focus, directrixand axis of symmetry. Determine the type of conic by using the discriminant

28. Routine • Sketch the graph of -4x2+9y2-48x-72y+144 = 0 Goal Problems: Determine choices for practice Recall & Reproduction

29. Non-Routine • 1. • 2. Write an equation for the ellipse with vertices (4, 0) and (–2, 0) and foci (3, 0) and (–1, 0).

30. Hyperbolas: For Both: center: (h, k) OR a a 0 vertices: (h , k – a) and (h, k + a) foci: (h, k – c) and (h, k + c) asymptotes: vertices: (h – a, k) and (h + a, k) foci: (h – c, k) and (h + c, k) asymptotes:

31. Example: Graph the following function:

32. Example: Graph the following function:

33. Routine Goal Problems: Determine choices for practice Recall & Reproduction

34. Non-Routine

35. Active Practice: