1. Writing Quadratic Equations when Given Vertex and Focus/ Directrix.
Vanessa Ledford
Algebra ll

2. Definition of Focus:
The parabola focus is a point from which distances are measured in forming a parabola and where these distances converge.
After changing the quadratic functions to vertex form, the formula for the Focus is (h, k+¼a)
“a” is the number that is in front of the parenthesis.
Vertex: (-1,-4)
Axis of symmetry: x = -1
Focus: (h, k+1/4a)
(-1,-4+¼( 1 ))
-4+ (-0.25)
= (-1,- 3.75)
f(x) = (x+1)2 – 4

3. Definition of Directrix:
A parabola Directrix is a line from which distances are measured in forming a parabola.
After changing the quadratic function in vertex form the formula for the Directrix is y = k – 1/4a
Vertex: (-1,-4)
Directrix: y = k – 1/4a
y = (-4) – ¼(1)
y = -4.25
f(x)= (x+1)2 – 4

4. Definition of the Axis of Symmetry:
The line perpendicular to the directrix and passing through the focus is called the axis of symmetry.

5. Steps to use: 1- Make a graph 2- Find the Vertex
3- Convert the quadratic function into Vertex Form
4- Next plug in the numbers for the focus or directrix while using the equation that applies to each of them.
Ex: If the focus is given then use (h, k+¼a), or if given the directrix use y = k – 1/4a. In certain cases if your given both the focus and directrix it doesn’t matter which equation to use.
5- Find the equation to use for finding the leading coefficient -“a”
6- Then solve for “a”
7- Finally write the Vertex Form

6. Showing a problem using formula for Focus:
Write the equation of the quadratic function with a vertex at (0,1) and focus at (0,4).
2. Vertex = (0,1)
6. 1+ (1/4a) = 4
(1/4a) = 3
(4a/1) * (1/4a) = 3 * (4a/1)
1 = 12a
(1/12)= (12a)/12
a = 1/12 1.
3. f(x) = a(x – 0)2 +1
a(x)2 +1
4. (h, k + 1/4a) = (0,4)
(0, 1 + 1/4a) = (0,4)
/4a = 4
7. f(x) = 1/12(x)2 + 1

7. Showing a problem using formula for Directrix:
Write the equation of the quadratic function with vertex at (1,8) and directrix at y = 3.
2. Vertex = (1, 8)
6. 3 = 8- (1/4)a
-5 = (-1/4a)
(-4a/1) *-5 = (-1/4a) * (-4a/1)
1 = 20a
1/20= 20a/20
a = 1/20 1.
3. f(x) = a(x-1)2 + 8
4. y = k – 1/4a
y = 3
5. 3 = 8- 1/4a
7. f(x) = 1/20 (x -1)2 +8

8. Example 1:
Write the equation of the quadratic function with a vertex at (3, -5) and focus at (3, -4.25).
/4a) = -4.25
(1/4a) = 0.75
(4a/1)* (1/4a) = 0.75 * (4a/1)
1= 3a
(1/3) = (3a)/3
a= 1/3 1.
2. Vertex = (3, -5)
3. f(x) = a (x- 3)2 -5
4. (h, k + 1/4a) = (3, -4.25)
( 3, /4a) = (3, -4.25)
/4a = -4.25
7. f(x) = 1/3 (x-3)2 – 5

9. Example 2: 7. f(x) = -1/16(x – 4)2 + 6
Write the equation of the quadratic function with focus at (4, 2) and directrix at y = 10.
8/2 = 4
/4a = 2
(1/4a) = -4
4a/1 * (1/4a) = -4 * 4a/1
1 = -16a
(-1/16) = -16a/-16
a = -1/16 1.
2. Vertex = (4, 6)
3. f(x) = a (x – 1/4a) = (4,2)
4. (h, k = 1/4a) = (4,2)
(4, 6 + 1/4a) = (4,2)
/4a = 2
7. f(x) = -1/16(x – 4)2 + 6

10. Example 3:
Write the equation of the quadratic function with focus at (6, 5) and directrix at y = -2.
7/-2 = -3.5
6. -2 = 1.5 – 1/4a
-3.5 = -1/4a
(-4a/1) * -3.5 = (-1/4a) * (-4a/1)
1 = 14a
(1/14) = (14a)/14
a = 1/14 1.
2. Vertex = (6, 1.5)
3. f(x) = a (x – 6)2 +1.5
4. y = k – 1/4a
y = -2
5. -2 = 1.5 – 1/4a
7. f(x) = 1/14(x – 6)2 +1.5

11. Work Cited
3.6 Focus and Directrix – Writing Quadratics from Info Part ll
Home Work. 3.6 Focus and Directrix