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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

Writing Equations of Translated Conic Sections

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Conic Section

Standard Form of Equation

Parabola Vertex (0,0)y = ax2

x = ay2

Vertex (h,k)y-k=a(x-h)2 or y=a(x-h)2+kx-k=a(y-h)2 or x=a(y-h)2+k

Conic Section

Standard Form of Equation

Parabola Vertex (0,0)y = ax2

x = ay2

Vertex (h,k)y-k=a(x-h)2 or y=a(x-h)2+kx-k=a(y-h)2 or x=a(y-h)2+k

Circle Center (0,0)x2 + y2 = r2

Center (h,k)(x-h)2 + (y-k)2 = r2

Conic Section

Standard Form of Equation

Parabola Vertex (0,0)y = ax2

x = ay2

Vertex (h,k)y-k=a(x-h)2 or y=a(x-h)2+kx-k=a(y-h)2 or x=a(y-h)2+k

Circle Center (0,0)x2 + y2 = r2

Center (h,k)(x-h)2 + (y-k)2 = r2

Ellipse Center (0,0)x2/a2 + y2/b2 = 1x2/b2 + y2/a2 = 1

Center (h,k)(x-h)2/a2 + (y-k)2/b2 = 1(x-h)2/b2 + (y-k)2/a2 = 1

Conic Section

Standard Form of Equation

Parabola Vertex (0,0)y = ax2

x = ay2

Vertex (h,k)y-k=a(x-h)2 or y=a(x-h)2+kx-k=a(y-h)2 or x=a(y-h)2+k

Circle Center (0,0)x2 + y2 = r2

Center (h,k)(x-h)2 + (y-k)2 = r2

Ellipse Center (0,0)x2/a2 + y2/b2 = 1x2/b2 + y2/a2 = 1

Center (h,k)(x-h)2/a2 + (y-k)2/b2 = 1(x-h)2/b2 + (y-k)2/a2 = 1

Hyperbola Center (0,0)x2/a2 - y2/b2 = 1y2/b2 - x2/a2 = 1

Center (h,k)(x-h)2/a2 - (y-k)2/b2 = 1(y-k)2/b2 + (x-h)2/a2 = 1

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. x2/4 – y2/16 = 1 x2 + y2 = 9 x2/4 + y2/16 = 1

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Example 2

Identify the conic section with equation 4x2 + y2 – 24x + 6y + 9 = 0

Identify the conic section represented by each equation: x2 + 14x – 4y + 29 = 0 x2 + y2 – 12x + 4y = 8

Describe the translation that would produce the equationx2 – 2y2 + 6x – 7 = 0

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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.