Skip to main content
Search IntMath

450+ Math Lessons written by Math Professors and Teachers

5 Million+ Students Helped Each Year

1200+ Articles Written by Math Educators and Enthusiasts

Simplifying and Teaching Math for Over 23 Years

Parabola - Interactive Graphs

You can explore various parabolas on this page, and see the effect of changing parameters (by dragging various points around).

For background information on what's going on, and more explanation, see:

The Parabola

Interactive Graph - Directrix and Focus of a Parabola

You can explore the concept of directrix and focus of a parabola in the following JSXGraph (it's not a fixed image).

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.

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

Sometimes the explanation boxes overlap. Sorry, but it can't be helped!

If the graph disappears, you can always refresh the page.

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)`.

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)` [This is the form given earlier in The Parabola page. I haven't multiplied it out so you can see the values.]
  2. `y = 1/(4p)(x − h)^2+ k` [This is in the easier to understand form, y = f(x).]

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 (yk)2 = 4(p)(xh), where the vertex is at (h, k) and p is the focal length. [This is the form given earlier in The Parabola page. Once again, I haven't multiplied out anything so you can see better where the values are coming from.]

Go back to « The Parabola.

Also, don't miss: How to draw y^2 = x − 2?, which has an extensive explanation of how to draw parabola graphs, depending on the formula given.