A parabola is the curve formed by all the points that are equally far from one point (the focus) and one line (the directrix).
This single 'equal distance' rule is what creates every property of the parabola.
It is also why satellite dishes and car headlights are shaped like parabolas: they gather light at the focus and send it back out in parallel rays.
Drag the focal distance p slider and the parabola widens and narrows, and the distance from a point P to the focus equals its distance to the directrix.
Intuition of Focus and Directrix
A quadratic graph and the focus-directrix rule name the same curve, but they open different doors. A parabola is the set of points P whose distance to the focus F equals the perpendicular distance to the directrix, and the standard form y² = 4px packs that rule into one coefficient. The focus is (p, 0), the directrix is x = −p, and the vertex is the origin, so reading 4p gives all three seats at once. The upright form is x² = 4py, and if the vertex shifts to (h, k) the focus shifts the same way. A bundle of parallel rays is a consequence of that equal-distance rule, not a property that appears merely because the equation is quadratic.
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Visualization of focus, directrix, and vertex of a parabola
The first sketch is the sideways curve y² = 4px. A gold point F sits at (p, 0), a blue dashed directrix stands on x = −p, and vertex O is at the origin. From the green point P a segment runs to F and another meets the directrix at a right angle, with PF = Pd written beside them. Focal distance p runs from 0.5 to 3 and starts at 1.5. A larger p places F farther right, the directrix farther left, and the curve wider. The second sketch overlays three comparison curves at p = 0.5, 1.0, and 2.0; the stroke nearest the chosen p is darker. The caption under that sketch states that a larger p makes a wider parabola.
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👀 What is a Parabola?
①Pick any point P on the parabola
②The distance to focus F equals the distance to the directrix
③This 'equal distance' condition defines a parabola
④Light from the focus reflects off the parabola in parallel rays — principle of satellite dishes and headlights!
Shape Change with p
If you read the coefficient on the right of y² = 4px as p itself, the focus and the directrix land four times too far. The number to match is 4p; the focus is (p, 0) and the directrix is x = −p. Thinking a larger p pinches the curve fights the second sketch. p = 2.0 opens wider than p = 0.5. Looking only at the vertex and skipping focus and directrix leaves you no check that the two lengths from a point on the curve must match. The p slider on this figure starts at 0.5, so only the right-opening shape is drawn; the left-opening case p < 0 is written as a sign in the formula.
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Parabola shape comparison for different p values
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📏 Meaning of p
①Large p: focus is far, parabola spreads wide
②Small p: focus is near, parabola is narrow
③The directrix is always at equal distance on the opposite side of the focus
④Compared curves are 0.5·1.0·2.0 only. Same p as Step 1. First 1.5 highlights none
Deriving the Standard Form
Standard Form (Horizontal)
y² = 4px
Focus F(p, 0), directrix x = −p, vertex at origin
Standard Form (Vertical)
x² = 4py
Focus F(0, p), directrix y = −p, vertex at origin
🔍 Derivation Steps
①Definition: PF = Pd (distance to focus = distance to directrix)
②From P(x,y) to F(p,0): √((x−p)² + y²)
③Distance to directrix x=−p: |x+p|
④Squaring both sides: (x−p)² + y² = (x+p)²
⑤Expanding gives y² = 4px. The sign of 4p is the opening direction
When Vertex Is Not at Origin
Translated Parabola
(y − k)² = 4p(x − h)
Vertex (h, k), focus (h+p, k)
🎯 Translation Essentials
①Replace x with (x−h), y with (y−k)
②Focus shifts by (h,k) in the same direction
③Directrix becomes x = h−p
④Formula structure is unchanged
Work It Out
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
In the form y²=4px, comparing the value of 4p gives the focus and directrix at once.
Example 2
Find the coordinates of the focus of the parabola (y−1)²=8(x−2).
1
The vertex is (2, 1) and from 4p=8 we get p=2.
4p=8 ⇒ p=2, vertex (2, 1)
2
The focus is at (h+p, k), shifted from the vertex by p along x.
focus (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.