• home
• sitemap
• LiveMath info
• Flash highlights
• Scientific Notebook
• math blog
• about
• feedback

4. The Parabola

Why study the parabola?

The parabola has many applications in situations where:

• Radiation needs to be concentrated at one point (e.g. radio telescopes, pay TV dishes, solar radiation collectors) or
• Radiation needs to be transmitted from a single point into a wide parallel beam (e.g. headlight reflectors).

Here is an animation showing how parallel radio waves are collected by a parabolic antenna.

Definition of a Parabola

The parabola is defined as the locus of a point which moves so that it is always the same distance from a fixed point (called the focus) and a given line (called the directrix).

In the following graph,

• The focus of the parabola is at (0, p).
• The directrix is the line y = -p.
• The focal distance is |p| (Distance from the origin to the focus, and from the origin to the directrix. We take absolute value because distance is positive.)
• The point (x, y) represents any point on the curve.
• The distance d from any point (x, y) to the focus (0, p) is the same as the distance from (x, y) to the directrix.

[The word locus means the set of points satisfying a given condition.
See some background in Distance from a Point to a Line.]

The Formula for a Parabola - Vertical Axis

Adding to our diagram from above, we see that the distance d = y + p.

Now, using the distance formula on the general points (0, p) and (x, y), and equating it to our value d = y + p, we have

Squaring both sides gives:

(x − 0)² + (y − p)² = (y + p)²

Simplifying gives us the formula for a parabola:

x² = 4py

In more familiar form, with "y = " on the left, we can write this as:

where p is the focal distance of the parabola.

---

Let's play with this in LiveMath. You can change the focal distance to see the effect it has on the parabola's shape.

LIVEMath

---

Now lets animate this, for positive values of p.

LIVEMath

---

Now let's see what "the locus of points equidistant from a point to a line" means in this LiveMath interactive.

LIVEMath

The LiveMath graph is similar to the following.

Each of the colour-coded line segments is the same length in this spider-like graph:

Example - Parabola with Vertical Axis

Sketch the parabola

Find the focal length and indicate the focus and the directrix on your graph.

Answer

Arch Bridges − Almost Parabolic

The Gladesville Bridge in Sydney, Australia was the longest single span concrete arched bridge in the world when it was constructed in 1964.

The shape of the arch is almost parabolic, as you can see in this image with a superimposed graph of y = −x² (The negative means the legs of the parabola face downwards.)

[Actually, such bridges are normally in the shape of a catenary, but that is beyond the scope of this chapter.]

Parabolas with Horizontal Axis

We can also have the situation where the axis of the parabola is horizontal:

In this case, we have the relation:

y² = 4px

[In a relation, there are two or more values of y for each value of x. On the other hand, a function only has one value of y for each value of x.]

Example - Parabola with Horizontal Axis

Sketch the curve and find the equation of the parabola with focus (-2,0) and directrix x = 2.

Answer

Shifting the Vertex of a Parabola from the Origin

This is a similar concept to the case when we shifted the centre of a circle from the origin.

To shift the vertex of a parabola from (0, 0) to (h, k), each x in the equation becomes (x − h) and each y becomes (y − k).

So if the axis of a parabola is vertical, and the vertex is at (h, k), we have

(x − h)² = 4p(y − k)

Let's see what this means in LiveMath:

LIVEMath

If the axis of a parabola is horizontal, and the vertex is at (h, k), the equation becomes

(y − k)² = 4p(x − h)

Exercises

1. Sketch x² = 14y

Answer

---

2. Find the equation of the parabola having vertex (0,0), axis along the x-axis and passing through (2,-1).

Answer

---

3. We found above that the equation of the parabola with vertex (h, k) and axis parallel to the y-axis is

(x − h)² = 4p(y − k).

Sketch the parabola for which (h, k) is (-1,2) and p = -3.

Answer

See also: How to draw y² = x - 2?

Applications of Parabolas

Application 1 - Antennas

A parabolic antenna has a cross-section of width 12 m and depth of 2 m. Where should the receiver be placed for best reception?

Answer

Application 2 - Projectiles

A golf ball is dropped and a regular strobe light illustrates its motion as follows...

We observe that it is a parabola. (Well, very close).

What is the equation of the parabola that the golf ball is tracing out?

Answer

 

Need a break? Play a math game. Well, they all involve math... No, really!

Chapter Contents

• Plane Analytical Geometry
• 1. Basic Definitions
• 2. The Straight Line
• Perpendicular Distance from a Point to a Line
• 3. The Circle
• 4. The Parabola
• 5. The Ellipse
• 6. The Hyperbola
• 7. Polar Coordinates
• 8. Curves in Polar Coordinates
• Equi-angular Spiral

• Comments, Questions?

From the math blog...