Parabola - Interactive Graphs

Interactive Graph - Directrix and Focus of a Parabola

Quick background: The parabola below has focus at F, and point P is at any position on the parabola. Point Q is the foot of the perpendicular to the directrix through P.

We have:

Distance PF = distance PQ

This follows from the definition of a parabola.

Things to do

Drag the point F (the focus) up or down the y-axis to see the effect on the shape of the parabola.

Drag the point D up and down. This will move the Directrix line up or down. Observe the effect on the shape of the parabola.

Drag the point P on the parabola and observe the distance d from the focus to the curve is always the same as the distance from the point P to the directrix, at Q.

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5
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-5
-10
5
-5
x
y
D

Directrix

F
P
Q

d = 5.80

d = 5.80

You can move the graph up-down, left-right if you hold down the "Shift" key and then drag the graph.

Parabola with vertex not at the origin

The vertex of a parabola is the "pointy end".

In the graph below, point V is the vertex, and point F is the focus of the parabola.

You can drag the focus, F, left-right, or up-down to investigate the formula of a parabola where the vertex is not at the origin (0,0)\displaystyle{\left({0},{0}\right)}.

You can also drag the directrix up and down to see the effect on the equation of the parabola.

The equation is given in 2 forms, where the vertex is at (h, k) and the focal length is p:

  1. (x−h)2=4(p)(y−k)\displaystyle{\left({x}-{h}\right)}^{2}={4}{\left({p}\right)}{\left({y}-{k}\right)}
  2. y=14p(x−h)2+k\displaystyle{y}=\frac{1}{{{4}{p}}}{\left({x}-{h}\right)}^{2}+{k} [This is in the easier to understand form, y = f(x).]
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5
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-5
-10
5
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x
y
D

Directrix

F
V

Equation of parabola:
(x − 3.0)2 = 4(4.0)(y − 1), OR
y = 0.06(x − 3.0)2 + 1.0

(3.0, 1.0)

Parabola with horizontal axis

Let's now play with a parabola which has horizontal orientation. That means the axis (the line running through the center of the parabola) is rotated 90° clockwise, compared to the parabolas above.

You can:

  • Drag the focus (point F) to see how it affects the shape of the parabola and its formula
  • Drag point D which will move the directrix line left and right, to see how it affects the shape of the parabola and its formula

The equation given is in the form (y − k)2 = 4(p)(x − h), where the vertex is at (h, k) and p is the focal length.

 –  o  +  ←  ↓  ↑  → 
5
10
-5
-10
5
-5
x
y
D

Directrix

F
V

Equation of parabola:
(y + 3)2 = 4(3.0)(x − 1.0)

(1.0, -3.0)

 

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