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LESSON 7–4

LESSON 7–4. Rotations of Conic Sections. Five-Minute Check (over Lesson 7-3) TEKS 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

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LESSON 7–4

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  1. LESSON 7–4 Rotations of Conic Sections

  2. Five-Minute Check (over Lesson 7-3) TEKS 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 Lesson Menu

  3. A.B. C.D. 5-Minute Check 1

  4. A. B. C. D. 5-Minute Check 2

  5. A. B. C. D. Graph the hyperbola 4x2 – y2 + 32x + 6y + 39 = 0. 5-Minute Check 3

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

  7. Determine the eccentricity of the hyperbola given by 9y2 – 4x2 – 18y + 24x – 63 = 0. A. 0.555 B. 0.745 C. 1.180 D. 1.803 5-Minute Check 5

  8. Targeted TEKS P.3(F) Determine the conic section formed when a plane intersects a double-napped cone. P.3(G) Make connections between the locus definition of conic sections and their equations in rectangular coordinates. Mathematical Processes P.1(A), P.1(C)

  9. 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. Then/Now

  10. Key Concept 1

  11. Write an Equation in the xy-Plane Use θ = 90° to write x2 + 3xy – y2 = 3 in the xy-plane. Then identify the conic. Find the equations for x and y. x = x cos θ–y sin θ Rotation equationsfor x and y y = x sin θ + y cos θ = –y sin 90 = 1 and cos 90 = 0 = x Example 1

  12. Write an Equation in the xy-Plane Substitute into the original equation. x2 + 3xy – y2 = 3 (–y)2 + 3(–y)(x) + (x)2 = 3 (y)2– 3xy + (x)2 = 3 Example 1

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

  14. A. B. C. D. Use θ = 60° to write 4x2 + 6xy + 9y2 = 12 in the xy-plane. Then identify the conic. Example 1

  15. Key Concept 2

  16. Write an Equation in Standard Form Using a suitable angle of rotation for the conic with equation x2 – 4xy – 2y2 – 6 = 0, write the equation in standard form. The conic is a hyperbola because B2 – 4AC > 0. Find θ. Rotation of the axes A = 1, B = –4, and C = –2 Example 2

  17. –3 Write an Equation in Standard Form Example 2

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

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

  20. Write an Equation in Standard Form Substitute these values into the original equation. x2 – 4xy – 2y2 = 6 Example 2

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

  22. A. B. C. D. Example 2

  23. Key Concept 2

  24. 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. Example 3

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

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

  27. Write an Equation in the xy-Plane Combine like terms. Simplify. Answer:3x2 + 2xy + 3y2– 32 = 0 Example 3

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

  29. Graph a Conic Using Rotations 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. Example 4

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

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

  32. Graph a Conic Using Rotations Example 4

  33. Graph a Conic Using Rotations Example 4

  34. Graph a Conic Using Rotations 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: Example 4

  35. A. B. C. D. Example 4

  36. Graph a Conic in Standard Form Use a graphing calculator to graph the conic section given by 8x2 + 5xy – 4y2 = –2. 8x2 + 5xy– 4y2 = –2 Original equation 8x2 + 5xy– 4y2 + 2 = 0 Add 2 to each side. –4y2 + (5x)y + (8x2 + 2) = 0 y-terms in quadratic form Quadratic formula Multiply. Example 5

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

  38. A. B. C. D. Use a graphing calculator to graph the conic section given by 3x2– 6xy + 8y2 + 4x– 2y = 0. Example 5

  39. LESSON 7–4 Rotations of Conic Sections

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