Graphing Conic Sections

Types of Conic Sections

1. Circle
2. Ellipse
3. Parabola
4. Hyperbola

Graphing Conic Sections

Circles

• General Equation: ( (x - h)^2 + (y - k)^2 = R^2 )
  • Center: ((h, k))
  • Radius: (R)
• Steps to Graph:
  1. Identify the center ((h, k)).
  2. Calculate the radius (R).
  3. Plot the center and use the radius to mark points on all directions (up, down, left, right).
  4. Connect these points to form a circle.

Ellipses

• General Equation:
  • ( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 ) or vice versa
  • (a) is the semi-major axis (larger one), (b) is the semi-minor axis.
• Steps to Graph:
  1. Identify the center ((h, k)).
  2. Determine (a) and (b).
  3. Plot the center.
  4. Plot vertices along the axes (a) and (b).
  5. Connect these points to form an ellipse.
• Major Axis & Minor Axis:
  • Major axis length: (2a)
  • Minor axis length: (2b)
• Vertices: End points of the major axis.
• Foci (focus points): Use (c^2 = a^2 - b^2).

Parabolas

• Standard Forms:
  • Horizontal: (y^2 = 4px) (opens left/right)
  • Vertical: (x^2 = 4py) (opens up/down)
• Vertex Form:
  • Horizontal: ((y - k)^2 = 4p(x - h))
  • Vertical: ((x - h)^2 = 4p(y - k))
• Steps to Graph:
  1. Identify the vertex ((h, k)).
  2. Find (p) (distance from vertex to focus).
  3. Determine direction based on the sign of (p).
  4. Plot vertex, focus, and directrix.

Hyperbolas

• General Equation:
  • ( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 ) or vice versa
  • (a) and (b) are distances from center to vertices and covertices respectively.
• Steps to Graph:
  1. Identify the center ((h, k)).
  2. Determine (a) and (b).
  3. Plot vertices and draw a rectangle (asymptote box).
  4. Draw asymptotes through the box diagonally.
  5. Sketch hyperbola opening along the axis of the first term.
• Vertices: End points of the transverse axis.
• Foci (focus points): Use (c^2 = a^2 + b^2).
• Asymptotes Equations:
  • (y - k = \pm \frac{b}{a}(x - h)) for horizontal opening
  • (y - k = \pm \frac{a}{b}(x - h)) for vertical opening

Identifying Conic Sections from Equations

1. Circle: Coefficients of (x^2) and (y^2) are the same and both positive.
2. Ellipse: Coefficients of (x^2) and (y^2) are positive but different.
3. Parabola: Only one squared term (either (x^2) or (y^2)).
4. Hyperbola: One squared term is positive, and the other is negative.

Standard Form Conversion

• Circles and Ellipses: Arrange terms, complete the square, and adjust the constant.
• Hyperbolas: Similar process but ensure the constant equals 1.
• Parabolas: Isolate squared term, complete the square for the other variable.

Conclusion

• Review general equations for each conic section.
• Practice by plotting simple examples.
• Use transformations to adjust standard forms.
• Understand characteristics: center, vertices, foci, directrix, and asymptotes for different conic sections.