This document provides information on translating and graphing different types of conic sections including parabolas, circles, ellipses, and hyperbolas. It discusses writing equations in standard form using properties like the vertex, center, foci, and distance from the focus to the vertex. The document also describes how to classify conic sections based on the discriminant of the equation and provides examples of classifying and graphing various conic sections.

2. Translated Parabolas
 Standard form – (h, k) is the vertex
Horizontal Axis Vertical Axis
Writing an Equation:
 Make a sketch to determine direction.
 Plug in h and k (from vertex).
 Distance between vertex and focus is p.
 Opens up or right: p is positive
 Opens down or left: p is negative

3. Example:
 Write an equation of the parabola with vertex
(-2, 1) and focus (-3, 1).

4. Translated Circles
 Standard form:
 Center is (h, k).
Graphing:
 Plot the center.
 Plot 4 points r units away.
 Draw the circle.

5. Example:
 Graph

6. Translated Ellipses
 Standard form – (h, k) is the center
Horizontal Major Axis Vertical Major Axis
Writing an Equation:
 Make a sketch.
 Find center – halfway between vertices.
 a is distance from vertex to center.
 c is from focus to center.

7. Example:
 Write an equation of the ellipse with foci at
(3, 5) and (3, -1) and vertices at (3, 6) and
(3, -2).

8. Translated Hyperbolas
 Standard form – (h, k) is the center
Horizontal Axis Vertical Axis
Graphing:
 Plot the center (h, k).
 Then graph as before.

9. Example:
 Graph

10. Classifying Conics
 Any conic can be written in the form:
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0
 The discriminant, B2 – 4AC, is used to classify.
Discriminantis: Type of Conic
< 0, B = 0 and A = C Circle
< 0, and either B ≠ 0 or A ≠ C Ellipse
= 0 Parabola
> 0 Hyperbola

11. To Graph:
 Classify first using discriminant.
 Write in standard form.
 Use Complete The Square
 Graph as before.

12. Example:
 Classify and graph the conic given by
2x2 + y2 – 4x – 4 = 0

13. Example:
 Classify and graph the conic given by
4x2 – 9y2 + 32x – 144y – 548 = 0

14. Your Turn!
 Classify and graph the conic given by
2x2 + y2 + 8y + 6 = 0