1. Splash Screen

2. Lesson Menu Five-Minute Check (over Lesson 7-3) Then/Now Key Concept:Rotation of Axes of Conics Example 1:Write an Equation in the x′y′-Plane Key Concept:Angle of Rotation Used to Eliminate xy-Term Example 2:Write an Equation in Standard Form Key Concept:Rotation of Axes of Conics Example 3:Real World Example: Write an Equation in the xy-Plane Example 4:Graph a Conic Using Rotations Example 5:Graph a Conic in Standard Form

3. Over Lesson 7-3 5-Minute Check 1 A.B. C.D.

4. Over Lesson 7-3 5-Minute Check 2 A.B. C.D.

5. Over Lesson 7-3 5-Minute Check 3 Graph the hyperbola 4x 2 – y 2 + 32x + 6y + 39 = 0. A.B. C.D.

6. Over Lesson 7-3 Write an equation for the hyperbola with foci (10, –2) and (–2, –2) and transverse axis length 8. 5-Minute Check 4 A. B. C. D.

7. Over Lesson 7-3 5-Minute Check 5 Determine the eccentricity of the hyperbola given by 9y 2 – 4x 2 – 18y + 24x – 63 = 0. A.0.555 B.0.745 C.1.180 D.1.803

8. Then/Now You identified and graphed conic sections. (Lessons 7–1 through 7–3) Find rotation of axes to write equations of rotated conic sections. Graph rotated conic sections.

9. Key Concept 1

10. Example 1 Use θ = 90° to write x 2 + 3xy – y 2 = 3 in the x y -plane. Then identify the conic. Find the equations for x and y. Write an Equation in the x y -Plane = –y x = x cos θ – y sin θ Rotation equations for x and y y = x sin θ + y cos θ sin 90 = 1 and cos 90 = 0 = x

11. Example 1 Substitute into the original equation. x 2 + 3xy – y 2 =3 (–y ) 2 + 3(–y )(x ) + (x ) 2 =3 (y ) 2 – 3x y + (x ) 2 =3 Write an Equation in the x y -Plane

12. Example 1 Answer: Write an Equation in the x y -Plane

13. Example 1 Use θ = 60° to write 4x 2 + 6xy + 9y 2 = 12 in the x y -plane. Then identify the conic. A. B. C. D.

14. Key Concept 2

15. Rotation of the axes Example 2 Write an Equation in Standard Form Using a suitable angle of rotation for the conic with equation x 2 – 4xy – 2y 2 – 6 = 0, write the equation in standard form. The conic is a hyperbola because B 2 – 4AC > 0. Find θ. A = 1, B = –4, and C = –2

16. Example 2 Write an Equation in Standard Form –3

17. Example 2 Write an Equation in Standard Form Use the half-angle identities to determine sin θ and cos θ. Half-Angle Identities Simplify.

18. Example 2 Write an Equation in Standard Form Next, find the equations for x and y. Rotation equations for x and y Simplify.

19. Example 2 Write an Equation in Standard Form Substitute these values into the original equation. x 2 – 4xy – 2y 2 = 6

20. Example 2 Write an Equation in Standard Form Answer:

21. Example 2 A. B. C. D.

22. Key Concept 2

23. Example 3 Write an Equation in the xy-Plane Use the rotation formulas for x and y to find the equation of the rotated conic in the xy-plane.

24. Example 3 Write an Equation in the xy-Plane Substitute these values into the original equation. = x cos 45° + y sin 45°θ = 45°= y cos 45° – x sin 45° Rotation equations for x′ and y′

25. Example 3 Write an Equation in the xy-Plane Original equation Multiply each side by 16. Substitute. Simplify. 2(x′) 2 + (y′) 2 = 16

26. Example 3 Write an Equation in the xy-Plane Answer: 3x 2 + 2xy + 3y 2 – 32 = 0 Combine like terms. Simplify.

27. Example 3 A. B. C. D. ASTRONOMY A sensor on a satellite is modeled by after a 60° rotation. Find the equation for the sensor in the xy-plane.

28. Example 4 The equation represents an ellipse in standard form. Use the center (0, 0), vertices (–6, 0), (6, 0), and co-vertices (0, –3) and (0, 3) in the x′y′-plane to determine the corresponding points for the ellipse in the xy-plane. Graph a Conic Using Rotations

29. Example 4 Find the equations for x and y for  = 60°. Graph a Conic Using Rotations x= x cos  – y sin  Rotation equations y= x sin  + y cos  for x and y Use the equations to convert the xy-coordinates of the vertex into xy-coordinates.

30. Example 4 Graph a Conic Using Rotations = –3

31. Example 4 Graph a Conic Using Rotations

32. Example 4 Graph a Conic Using Rotations

33. Example 4 The new vertices and co-vertices can be used to sketch the ellipse. They can also be used to identify the x′y′-axis. Answer: Graph a Conic Using Rotations

34. Example 4 A.B. C.D.

35. Example 5 Use a graphing calculator to graph the conic section given by 8x 2 + 5xy – 4y 2 = –2. 8x 2 + 5xy – 4y 2 = –2Original equation 8x 2 + 5xy – 4y 2 + 2= 0Add 2 to each side. –4y 2 + (5x)y + (8x 2 + 2)= 0y-terms in quadratic form Graph a Conic in Standard Form Quadratic formula Multiply.

36. Example 5 Graph a Conic in Standard Form Simplify. Graphing both equations on the same screen yields the hyperbola. Answer:

37. Example 5 Use a graphing calculator to graph the conic section given by 3x 2 – 6xy + 8y 2 + 4x – 2y = 0. A.B. C.D.

38. End of the Lesson