A focus, a line, and equal distances.
That equal-distance rule is the definition. Everything else — the formulas, the shape, the reflector property — is a consequence of it.
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- The Parabola
The Parabola
Every point on it is the same distance from one point and one line.
Find the focus
x² = 12y
Compare with x² = 4cy: here 4c = 12, so c = 3. The x² term means the parabola opens up or down along the y-axis, so the focus is (0, 3) and the directrix is y = −3. A takes the 12 as c without dividing by 4. C puts the focus on the wrong axis — the squared variable tells you which way it opens.
The squared variable says which way it opens.
x² = 4cy
opens up if c > 0
focus (0, c), directrix y = −c
x² = −4cy
opens down
focus (0, −c)
y² = 4cx
opens right
focus (c, 0), directrix x = −c
x squared opens vertically, y squared opens sideways
It feels backwards at first. The trick is to ask which variable is free to take any value: in x² = 4cy, y can be anything positive, so the curve runs upward. The focus always sits on the axis the curve opens along.
Match the form, then read everything off.
y² = −20x matches y² = −4cx
y squared, so it opens sideways
4c = 20, so c = 5
always divide by 4
negative, so it opens LEFT
focus (−5, 0)
directrix x = 5
on the far side of the vertex
The focus is always inside the curve and the directrix always outside, the same distance the other way, so focus and directrix sit 2c apart (here 5 + 5 = 10). If your two answers end up on the same side, a sign has gone astray.
A shifted vertex works exactly like a shifted circle.
(x − h)² = 4c(y − k)
the general form
(x − 4)² = 12(y + 2)
h = 4, k = −2
vertex (4, −2), 4c = 12 so c = 3
opens upward
focus (4, 1), directrix y = −5
3 up and 3 down from the vertex
Find the vertex first, then step c units along the axis of symmetry for the focus and c units the other way for the directrix. Everything is measured from the vertex.
Variations: working backwards, a fractional c, and a general form.
Three versions of the same reading-off, each with one new wrinkle.
Backwards: the focus is (0, −3) and the directrix is y = 3. Find the equation.
vertex (0, 0)
halfway between focus and directrix
focus below the vertex, so it opens DOWN
c = 3
x² = −12y
4c = 12, with the minus for down
Here we build the equation from the picture. Sketching the focus and the line first settles the sign.
A fractional c: y² = 6x
4c = 6, so c = 64 = 32
divide by 4, even when it is not whole
focus (32, 0), directrix x = −32
opens right
latus rectum = 4c = 6
the coefficient itself
Nothing changes when c is a fraction. Keep it as 32; a rounded 1 or 2 puts the focus in the wrong place.
A general form: x² + 6x − 4y + 1 = 0
x² + 6x = 4y − 1
keep the x terms on the left
x² + 6x + 9 = 4y + 8
add 9 to both sides
(x + 3)² = 4(y + 2)
factor the 4 out of the right side
vertex (−3, −2), c = 1, focus (−3, −1)
directrix y = −3
Factor the coefficient out of the right side before reading k; 4y + 8 hides k = −2.
The latus rectum sets the width, and the shape does the work.
latus rectum
length 4c
the chord through the focus
large c
wide, shallow curve
focus far from the vertex
small c
narrow, steep curve
focus close to the vertex
Every ray coming in parallel to the axis bounces off a parabola straight through the focus. That is why satellite dishes, car headlamps and solar cookers are all parabolic — and why the focus has its name.
On the exam
Tests give an equation and ask for the focus, directrix, vertex or the way it opens, or give the focus and directrix and ask for the equation. The fastest route: see which variable is squared, divide the other coefficient by 4 to get c, and step c from the vertex. A 10-second check: the vertex must sit exactly halfway between the focus and the directrix.
Now you try
Practice problem
Find the focus and directrix
y² = 16x
4c = 16 gives c = 4, and y² means it opens sideways — to the right, since the sign is positive. So the focus is (4, 0) and the directrix is x = −4. A puts them on the wrong axis, and D halves the 16 instead of quartering it.
Hint: y is squared, so the curve opens along the x-axis. Divide 16 by 4, not by 2.