A parabola is the curve made of all points that are the same distance from a fixed point, the focus, and a fixed line, the directrix.
That single equal-distance condition gives the parabola its equation and all its properties.
Satellite dishes and car headlights are parabolic because light from the focus reflects off a parabola parallel to its axis.
Move the focal distance p and the position of point P: however wide the parabola gets, PF and PD always come out equal.
Focus and Directrix
A parabola first appears as the graph of a quadratic function, but defining it through a focus and a directrix explains in a single condition why the curve has its shape. Writing in coordinates the condition that a point P on the curve is as far from the focus F as from the directrix gives the standard form y² = 4px. Reading off 4p from this equation fixes the focus (p, 0), the directrix x = −p and the vertex at the origin all at once. A parabola that opens up or down has the form x² = 4py, and when the vertex moves to (h, k), the focus and the directrix move the same way by the same amount. The fact that light from the focus reflects into a beam parallel to the axis is also a consequence of this distance condition.
seegongsik.com
The focus F and directrix of y² = 4px, a point P on the curve and the foot D of its perpendicular
The first picture shows the right-opening parabola y² = 4px. The gold point is the focus F(p, 0), the blue dashed line is the directrix x = −p, and the green point P moves along the curve. The green segment is PF and the blue segment is PD, the perpendicular from P to the directrix; the top band shows both lengths. Move P up or down, or change the focal distance p anywhere from 0.5 to 3, and PF and PD stay equal, both equal to x + p. The second picture has faint reference curves for p = 0.5, 1 and 2, with the curve and focus for the chosen p drawn over them in gold. Compared with the reference curves, it is clear that a larger p makes the curve open wider.
1.5
2.5
👀 Definition of a parabola
①Wherever P sits on the curve, PF = PD
②This equal-distance condition is the definition of a parabola
③Since PD = x + p, PF is also x + p, and writing this out as an equation gives y² = 4px
④Light from the focus reflects off the parabola parallel to its axis, which is how satellite dishes and headlights work
How p Changes the Shape
Reading the coefficient of x in y² = 4px as p instead of 4p puts the focus and the directrix four times too far out. For y² = 12x, 4p = 12, so p = 3, the focus is (3, 0) and the directrix is x = −3. It is tempting to think a larger p makes the curve narrower, but it is the other way round. At the same x, the size of y in y² = 4px grows with p, so in the second picture the curve with p = 2 opens wider than the one with p = 0.5. And if you look only at the vertex and forget the focus and directrix, you lose the check that the two distances from a point on the curve are equal. The picture shows only p > 0; with p < 0 the same equation describes a parabola opening to the left.
seegongsik.com
Reference curves for p = 0.5, 1 and 2 and the parabola for the chosen p
1.5
📏 What p means
①p is the distance from the vertex to the focus, and also from the vertex to the directrix
②A larger p puts the focus farther away and opens the parabola wider
③A smaller p brings the focus close to the vertex and makes the parabola narrower
④The chord through the focus perpendicular to the axis has length 4p
Deriving the Standard Form
Standard form (opening left or right)
y² = 4px
Focus F(p, 0), directrix x = −p, vertex at the origin
Standard form (opening up or down)
x² = 4py
Focus F(0, p), directrix y = −p, vertex at the origin
🔍 Setting up the equation
①The definition is PF = PD
②The distance from P(x, y) to F(p, 0) is √((x − p)² + y²)
③The distance to the directrix x = −p is |x + p|
④Setting them equal and squaring gives (x − p)² + y² = (x + p)²
⑤Simplifying gives y² = 4px, and the sign of p decides which way it opens
When the Vertex Is Not the Origin
Translated parabola
(y − k)² = 4p(x − h)
Vertex (h, k), focus (h + p, k), directrix x = h − p
🎯 The key to translation
①Replace x with x − h and y with y − k
②The focus moves the same way, to (h + p, k)
③The directrix moves to x = h − p
④The relationship between p and 4p is unchanged
Worked Examples
Example 1
Find the focus and directrix of the parabola y² = 12x.
1
Compare with the standard form y² = 4px to find p.
4p = 12 ⇒ p = 3
2
The focus is (p, 0) and the directrix is x = −p.
focus (3, 0), directrix x = −3
▸ Focus (3, 0), directrix x = −3
For y² = 4px, comparing only the value of 4p gives the focus and directrix straight away.
Example 2
Find the coordinates of the focus of the parabola (y − 1)² = 8(x − 2).
1
The vertex is (2, 1), and 4p = 8 gives p = 2.
4p = 8 ⇒ p = 2, vertex (2, 1)
2
The focus is the vertex moved p in the x-direction, (h + p, k).
(2 + 2, 1) = (4, 1)
▸ (4, 1)
For a translated parabola, add p to the vertex (h, k) to get the focus. The directrix is x = h − p = 0.
Summary
Key formula
y² = 4px
Focus (p, 0), directrix x = −p; opens to the right if p > 0 and to the left if p < 0
2023 KICE mock exam Math (Geometry) type, adapted
If the focus of the parabola y² = 4px is (2, 0), what is the equation of the directrix?
①x = −2
②x = 2
③x = −1
④x = 1
⑤y = −2
▸ ① x = −2
1
The focus of y² = 4px is (p, 0), so p = 2.
2
The directrix is x = −p.
directrix x = −2
🎯 Exam Points
①A parabola is defined by equal distances to the focus and the directrix
②In y² = 4px, the sign of 4p tells you which way it opens
③Read 4p from the coefficient of x, find p, then write the focus and directrix
④If the vertex is (h, k), think of a translation
⑤The distance from a point on the curve to the focus is easier to compute as the distance to the directrix