Finding the Center, Foci, and Directrix of a Parabola

Overview

In this lecture, we focus on finding the center, foci, and directrix of a parabola in standard form.

Steps to Determine the Characteristics of the Parabola

1. Determine Orientation

   • Identify whether the parabola is vertical or horizontal.
   • In this case, it's a vertical parabola.

2. Find the Center

   • The center is given by the formula:

     Center: ( (h, k) )

   • For this problem, the center is:

     • ( h = -3 )
     • ( k = 1 )
     • Therefore, Center: ( (-3, 1) )

3. Determine the Focus

   • The focus for a vertical parabola is given by:

     Focus: ( (h, k + p) )

   • To find ( p ):

     • The equation is of the form:

       ( x - h = 4p(y - k) )

     • From the problem, we have:

       • ( -2 = 4p )

     • Solving for ( p ):

       • ( p = -\frac{1}{2} )

   • Plugging in the values:

     • Focus: ( (-3, 1 - \frac{1}{2}) = (-3, \frac{1}{2}) )

4. Find the Directrix

   • The directrix is calculated using:

     Directrix: ( y = k - p )

   • Plugging in the values:

     • Directrix:
       • ( y = 1 - (-\frac{1}{2}) )
       • ( y = 1 + \frac{1}{2} )
       • ( y = \frac{3}{2} )

   • Note: The directrix is a line, while the focus is a point.

Key Takeaways

• For vertical parabolas:
  • Center: ( (h, k) )
  • Focus: ( (h, k + p) )
  • Directrix: ( y = k - p )
• Remember to differentiate between adding and subtracting for finding focus and directrix based on orientation.
• Horizontal parabolas would involve adjusting ( h ) instead of ( k ) for focus and directrix calculations.