https://file.aiquickdraw.com/mathbot/file/17288031592251p526rgy.jpg

Show a clear sketch of each graphs before and after the transformation

Here is the sketch showing the graphs of the three conic sections before and after the translation:

- (a) Parabola with the original vertex at (3, 2), translated to the origin.
- (b) Ellipse centered at (-3, 0), translated to the origin.
- (c) Parabola with the original vertex at (2, 1), translated to the origin.

If you need further clarifications or adjustments, let me know!

Here are five related questions for deeper understanding:
1. How does the general form of a conic section equation change after translation?
2. What is the significance of completing the square when handling conic sections?
3. How do you determine the direction of opening for a translated parabola?
4. What are the axes lengths of an ellipse, and how do they relate to the equation?
5. How can we classify different conic sections based on their equations?

**Tip:** Completing the square is often essential for identifying the key properties (e.g., vertex, center) of conic sections, especially in non-standard forms.

Translate so that the center or vertex of the conic section is at the origin of the new coordinate system. Sketch the curve showing both the old and new axes. a) (x - 3)^2 = 8(y - 2) b) 16(x + 3)^2 + 25y^2 = 400 c) y^2 - 4x - 2y + 9 = 0

Learn how to translate conic sections so that their centers or vertices are at the origin, including parabola and ellipse transformations. This problem covers the translation of axes, key conic section formulas, and includes example sketches of the curves before and after translation.

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