4. The Parabola
Why study the parabola?
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The parabola has many applications in situations where:
- Radiation often 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. The parallel rays reflect off the antenna and meet at a point (the red dot, labelled F), called the focus.
Click the "See more" button to see more examples. Each time you run it, the dish will become flatter.
Observe that the focus point, F, moves further away from the dish each time you run it.
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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).
[The word locus means the set of points satisfying a given condition. See some background in Distance from a Point to a Line.]
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 axis of symmetry of this parabola is the y-axis.
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.
Now let's see what "the locus of points equidistant from a point to a line" means.
Each of the colour-coded line segments is the same length in this spider-like graph:
Don't miss Interactive Parabola Graphs, where you can explore concepts like focus, directrix and vertex.
Example - Parabola with Vertical Axis
Need Graph Paper?
Sketch the parabola
Find the focal length and indicate the focus and the directrix on your graph.
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. See Is the Gateway Arch a Parabola?]
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: (not function)
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.]
The above graph's axis of symmetry is the x-axis.
Example - Parabola with Horizontal Axis
Sketch the curve and find the equation of the parabola with focus (-2,0) and directrix x = 2.
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)
In the above case, the axis of symmetry is the vertical line through the point (h, k), that is x = h.
If the axis of a parabola is horizontal, and the vertex is at (h, k), the equation becomes
(y − k)2 = 4p(x − h)
In the above case, the axis of symmetry is the horizontal line through the point (h, k), that is y = k.
1. Sketch `x^2= 14y`
2. Find the equation of the parabola having vertex (0,0), axis along the x-axis and passing through (2,-1).
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`.
Helpful article and graph interactives
See also: How to draw y^2 = x − 2?, which has an extensive explanation of how to manipulate parabola graphs, depending on the formula given.
Also, don't miss Interactive Parabola Graphs, where you can explore parabolas by moving them around and changing parameters.
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?
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?
Conic section: Parabola
All of the graphs in this chapter are examples of conic sections. This means we can obtain each shape by slicing a cone at different angles.
How can we obtain a parabola from slicing a cone?
We start with a double cone (2 right circular cones placed apex to apex):
If we slice a cone parallel to the slant edge of the cone, the resulting shape is a parabola, as shown.