Lecture on Graphing Conic Sections

Introduction

• Focus on graphing conic sections: circles, ellipses, parabolas, and hyperbolas.
• Identify the type of conic section from an equation and convert it to standard form.

Graphing Circles

Standard Equation

• Circle equation: ( (x - h)^2 + (y - k)^2 = r^2 )
  • Center: ((h, k))
  • Radius: ( r )

Examples

1. Equation: (x^2 + y^2 = 9)

   • Center: (0, 0)
   • Radius: ( r = 3 )
   • Graph: Go 3 units in each direction from the origin and draw a circle.

2. Equation: ((x - 3)^2 + (y + 4)^2 = 16)

   • Center: (3, -4)
   • Radius: ( r = 4 )
   • Graph: Plot center and extend 4 units in each direction.

3. Equation: ((x + 2)^2 + (y - 3)^2 = 4)

   • Center: (-2, 3)
   • Radius: ( r = 2 )
   • Graph: Extend 2 units from center in each direction.

Graphing Ellipses

Standard Equation

• Ellipse equation: ( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 )
  • Center: ((h, k))
  • (a) is the larger value, determining the major axis.
  • Axis alignment determined by larger denominator.

Examples

1. Equation: ( \frac{x^2}{25} + \frac{y^2}{49} = 1 )

   • Center: (0, 0)
   • (a = 7, b = 5)
   • Major axis: Vertical (14 units), Minor axis: Horizontal (10 units)

2. Equation: ( \frac{(x - 2)^2}{16} + \frac{(y + 3)^2}{9} = 1 )

   • Center: (2, -3)
   • (a = 4, b = 3)
   • Major axis: Horizontal

Graphing Hyperbolas

Standard Equation

• Hyperbola equations:
  • ( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 )
  • ( \frac{(y - k)^2}{a^2} - \frac{(x - h)^2}{b^2} = 1 )
  • (a) is associated with the term in front.

Examples

1. Equation: ( \frac{x^2}{9} - \frac{y^2}{4} = 1 )
   • Center: (0, 0), Opens horizontally.
   • (a = 3, b = 2)
   • Asymptotes: (y = \pm \frac{2}{3}x)

Graphing Parabolas

Standard Equation

• Parabola equations:
  • Vertical: (x^2 = 4py)
  • Horizontal: (y^2 = 4px)
  • Vertex: ((h, k))
  • Focus: (p) units from the vertex.

Examples

1. Equation: (y^2 = 8x)
   • Vertex: (0, 0), Opens right.
   • (p = 2)
   • Focus: (2, 0), Directrix: (x = -2)

Identifying Conic Sections

• Parabola: Only one squared term.
• Hyperbola: One positive and one negative squared term.
• Ellipse: Both terms positive, different coefficients.
• Circle: Both terms positive, same coefficients.

Converting to Standard Form

Steps

1. Rearrange to group variables.
2. Complete the square.
3. Adjust constants to maintain equality.

Example Transformations

• Converting general equations to standard form and determining key properties like center, axis, radius, etc.

Review: General Equations

• Circle: ((x - h)^2 + (y - k)^2 = r^2)
• Ellipse: ( \frac{(x - h)^2}{a^2} + \frac{(y - k)^2}{b^2} = 1 )
• Hyperbola: ( \frac{(x - h)^2}{a^2} - \frac{(y - k)^2}{b^2} = 1 )
• Parabola: Depending on orientation, use appropriate form.

Conclusion

• Understanding key differences between conic sections and how to graph them effectively.