SME
Grade 12 Math · Problem type
Parabola: focus and directrix
Use this for “Find the focus and directrix of (x − 2)² = 8(y + 1).”
Diagram
(x − 2)² = 8(y + 1)
Vertex (2, −1)
focus (2, 1), p = 2 away
directrix y = −3.
Every point is as far from the focus as from the directrix.
The point (6, 1) is on the curve.
Focus, y: 1
Directrix, y: −3
This is an example. Type your own values to start a new problem.
(x
−
h
)²
=
q
(y
−
(
k
)
)
p
Vertex to focus
worked out (not typed)
F
Focus, y
calculated
L
Directrix, y
calculated
x
Point, x
example
y
Point, y
calculated
Formulas
p
=
q
4
2
=
8
4
F
=
k
+
p
1 = −1 + 2
L
=
k
−
p
−3 = −1 − 2
(
x
−
h
)
2
=
q
×
(
y
−
k
)
(6 − 2)
2
=
8
×
(1
−
(−1))
Assumptions
•
In (x − h)² = 4p(y − k), the vertex is (h, k) and the number before (y − k) is 4p.
•
The focus is p above the vertex and the directrix p below it (p < 0 opens down).
•
Every point on the parabola is as far from the focus as from the directrix.
Step-by-step
WE KNOW
x = 6, h = 2, k = −1, q = 8
FIND
vertex to focus (p), focus, y (F), directrix, y (L), point, y (y)
Step 1 · Find vertex to focus (p)
Use p = q ÷ 4
The
number
before
(
y
−
k
)
is
4
p
:
divide
it
by
4.
p
=
8
4
p
=
2
Step 2 · Find focus, y (F)
Use F = k + p
The
focus
is
p
from
the
vertex,
along
the
axis.
F
=
−1
+
2
F
=
1
Step 3 · Find directrix, y (L)
Use L = k − p
The
directrix
is
p
from
the
vertex
on
the
other
side.
L
=
−1
−
2
L
=
−3
Step 4 · Find point, y (y)
Use (x − h)² = q × (y − k)
Divide
both
sides
by
4
p
,
then
add
k
.
y
=
k
+
(
x
−
h
)
2
÷
q
y
=
−1
+
(6 − 2)
2
÷
8
y
=
−1
+
4
2
8
y
=
−1
+
16
8
y
=
−1
+
2
y
=
1
Check
Put the numbers back in. Both sides should match.
2
=
8
4
✓
1 = −1 + 2 ✓
−3 = −1 − 2 ✓
(6 − 2)
2
=
8
×
(1
−
(−1))
✓
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