Lecture Notes on Parabolas

Introduction to Parabolas

• Equations of Parabolas
  • Left-opening parabola: ( y^2 = 4px )
  • Right-opening parabola: ( x^2 = 4py )
  • Focus and directrix:
    • Focus: Point where the parabola opens towards
    • Directrix: Line perpendicular to the axis of symmetry, ( p ) units away from the vertex in the opposite direction

Key Concepts

• Parameter ( p ): Distance from vertex to focus and vertex to directrix

  • If ( p ) is positive, the parabola opens to the right/upward
  • If ( p ) is negative, it opens to the left/downward

• Latus Rectum: Line segment through the focus, perpendicular to the axis of symmetry, with length ( 4p )

Graphing Parabolas

1. Example: Graphing ( x^2 = 4py )

   • Calculate ( p ): Set ( 4p = 8 ), thus ( p = 2 )
   • Plot focus at ( (0, 2) ) and directrix at ( y = -2 )
   • Points plotted at ( 2p ) units to the left and right
   • Sketch the parabola

2. Finding Focus and Directrix

   • Given ( y^2 = 4px ):
     • Set ( 4p = 2 ), ( p = \frac{1}{2} )
     • Focus at ( (0.5, 0) ), directrix at ( x = -0.5 )
     • Length of latus rectum = 2

3. Matching Equations to Graphs

   • Positive coefficients for upwards/rightwards openings, negative for downwards/leftwards

4. Writing Standard Equations

   • Given focus and directrix:
     • Focus at ( (-3, 0) ), directrix at ( x = 3 )
     • Calculate vertex as midpoint: ( (0, 0) )
     • Determine ( p = -3 )
     • Resulting equation: ( y^2 = -12x )

5. Finding Vertex and Focus from Given Equation

   • Example: ( y^2 = 4px ) shifted to ( (h,k) )
   • Determine vertex, focus, directrix, length of latus rectum

6. Finding Domain and Range

   • For horizontal parabolas: Domain = all real numbers, Range = from vertex's y value
   • For vertical parabolas: Domain = from vertex's x value, Range = all real numbers

Conclusion

• Understanding the properties and equations of parabolas is essential for graphing and problem-solving. Ensure familiarity with how to derive and manipulate their equations.