1. 1 Section 10.6: Translating Conic Sections What You’ll Learn: to translate conic sections and write and identify the equation of translated conic section Why: to use location navigation systems

2. Writing Equations of Translated Conic Sections 2 Conic Section Standard Form of Equation ParabolaVertex (0,0) y = ax 2 x = ay 2 Vertex (h,k) y-k=a(x-h) 2 or y=a(x-h) 2 +k x-k=a(y-h) 2 or x=a(y-h) 2 +k Conic Section Standard Form of Equation ParabolaVertex (0,0) y = ax 2 x = ay 2 Vertex (h,k) y-k=a(x-h) 2 or y=a(x-h) 2 +k x-k=a(y-h) 2 or x=a(y-h) 2 +k CircleCenter (0,0) x 2 + y 2 = r 2 Center (h,k) (x-h) 2 + (y-k) 2 = r 2 Conic Section Standard Form of Equation ParabolaVertex (0,0) y = ax 2 x = ay 2 Vertex (h,k) y-k=a(x-h) 2 or y=a(x-h) 2 +k x-k=a(y-h) 2 or x=a(y-h) 2 +k CircleCenter (0,0) x 2 + y 2 = r 2 Center (h,k) (x-h) 2 + (y-k) 2 = r 2 EllipseCenter (0,0) x 2 /a 2 + y 2 /b 2 = 1 x 2 /b 2 + y 2 /a 2 = 1 Center (h,k) (x-h) 2 /a 2 + (y-k) 2 /b 2 = 1 (x-h) 2 /b 2 + (y-k) 2 /a 2 = 1 Conic Section Standard Form of Equation ParabolaVertex (0,0) y = ax 2 x = ay 2 Vertex (h,k) y-k=a(x-h) 2 or y=a(x-h) 2 +k x-k=a(y-h) 2 or x=a(y-h) 2 +k CircleCenter (0,0) x 2 + y 2 = r 2 Center (h,k) (x-h) 2 + (y-k) 2 = r 2 EllipseCenter (0,0) x 2 /a 2 + y 2 /b 2 = 1 x 2 /b 2 + y 2 /a 2 = 1 Center (h,k) (x-h) 2 /a 2 + (y-k) 2 /b 2 = 1 (x-h) 2 /b 2 + (y-k) 2 /a 2 = 1 HyperbolaCenter (0,0) x 2 /a 2 - y 2 /b 2 = 1 y 2 /b 2 - x 2 /a 2 = 1 Center (h,k) (x-h) 2 /a 2 - (y-k) 2 /b 2 = 1 (y-k) 2 /b 2 + (x-h) 2 /a 2 = 1

3. Example 1  Write the equation of each conic section: Ellipse with center (-3,-2); vertical major axis of length 8; minor axis of length 6. Hyperbola with vertices (0,1) and (6,1) and foci (-1,1) and (7,1)  Translate each of the following situations 3 units up and 2 units left. x 2 /4 – y 2 /16 = 1 x 2 + y 2 = 9 x 2 /4 + y 2 /16 = 1 3

4. Example 2  Identify the conic section with equation 4x 2 + y 2 – 24x + 6y + 9 = 0  Identify the conic section represented by each equation: x 2 + 14x – 4y + 29 = 0 x 2 + y 2 – 12x + 4y = 8  Describe the translation that would produce the equation x 2 – 2y 2 + 6x – 7 = 0

5. 5 Homework Section 10-6 HW pages 495-496: 1-21 (no graphs) Any questions that involve writing or explaining should be done in complete sentences and show critical thinking skills.