What is a parabola, and of what use is it?
If a sheet of paper is likened to an infinite plane in space, the x- and y-axes, drawn at right angles to each other, provide a means of describing each point on the plane of the paper in terms of an x and a y value. Thus the point (1, 3) tells us that beginning at the origin or center where the x-axis crosses the y-axis, if we travel one unit to the right and then three units up, we will have reached the point we seek.
Introducing the Parabola
Now we will pass on to describing a parabola.¹ The parabola is an important mathematical “curve,” inasmuch as it describes, mathematically, the behavior of a number of important actions, such as the stretching of a spring. If the distance a spring is stretched is equal to the x-value, and the necessary force to accomplish this stretching is the y-value, the curve is a parabola. The force needed to stretch the spring greatly increases the further we stretch it. In the same way, the y values quickly increase even though values of x don’t increase much.
As the equation for a line is
y = mx + b
where m is the slope of the line and b is its intercept, so the equation for a parabola centered on the y-axis is,
y = ax2 + bx + c
where a ≠ 0
If a = 0, the term ax2 becomes zero, and the equation reduces to y = c, a simple line equation.
In the case that a = 1, b = 0, and c = 0, we have a particularly simple parabola. y = x2. For that equation, we can draw up a list of points to help us draw the graph. For instance,
As is seen in the illustration, the parabola is shaped somewhat like the letter U, only with the branches of the U always getting slightly further apart. The bottom of the U intercepts the y-axis at b, even as was the case with the line. Since in this instance, b = 0, the bottom of the parabola is located at y = 0.
Practical Use of the Parabolic Shape
We’ve already seen how a parabolic curve describes the action of stretching a spring; however, there are many other practical applications utilizing the shape of a parabola. Parabolic mirrors are used in some telescopes and other optical devices. In addition, ideally the best shape to receive a signal is the parabolic dish.