Teaching Notes:

In this lesson, students look at the directrix and the focus of a parabola. This lesson has direct connections to Day 1. However, instead of finding the parabola from the directrix and focus, the students will be finding the directrix and focus of a known parabola. No technology is stressed in this lesson. However, algebraic manipulation tools may be helpful for some students that have difficulty performing tasks like completing the square.

Introduction:

Prior to this lesson, we have looked at the parabola's equation in standard polynomial form. That is, y = ax² + bx + c. Now we will look at simply the standard form of the parabola equation.

Standard Form Equation: (x - h)² = 4c(y - k)
Vertex: (h,k)
Axis of Symmetry: x = h
Focus: (h,k+c)
Directrix: y = k - c
Opens: up if c > 0 / down if c < 0

Examples:

1. Find the directrix and focus of this parabola:
   (x - 5)² = 12(y + 2)
   Solution:
   This equation is already in standard form, so simply use the information provided above.
   Value of "c": 3 Vertex: (5,-2)
   Axis of Symmetry: x = 5
   Focus: (5,1)
   Directrix: y = -5
   Opens: up
   
2. Find the directrix and focus of this parabola:
   2x² - 4x + y + 4 = 0
   Solution:
   This equation needs to first be written in standard form by completing the square.
   2x² - 4x + y + 4 = 0
   2x² - 4x + 2 + y + 4 = 2 (add 2 to both sides)
   2(x² - 2x + 1) + (y + 4) = 2 (grouping and factoring)
   2(x - 1)² = -y - 2 (factoring)
   (x - 1)² = -(1/2)(y + 2)
   Value of "c": -1/8 Vertex: (1,-2)
   Axis of Symmetry: x = 1
   Focus: (1,-17/8)
   Directrix: y = -15/8
   Opens: down
   

Additional Teaching Notes:

Some students may require some additional examples. Probably the most difficulty will be encountered while "completing the square", so it may be necessary to spend some time reviewing this concept.

Homework Problems:

Find the directrix and focus of these parabola:

1. y = x² + 5
2. (x - 4)² = 2(y - 1)
3. 3x² + 12x - 5y + 7 = 0
4. y = x² - 6x + 5
5. y = -2x² + 5x - 3

Solutions to Homework Problems:

1. Focus: (0,-19/4) Directrix: y = -21/4
2. Focus: (4,3/2) Directrix: y = 1/2
3. Focus: (-2,-7/12) Directrix: y = -17/12
4. Focus: (3,-15/4) Directrix: y = -17/4
5. Focus: (5/4,0) Directrix: y = 1/4

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