Analytical Geometry - Lesson 2: The Parabola

Lesson 1 introduced us to the coordinate axes, and the equation for a line. If a sheet of paper is likened to an infinite plane in space, the x- and y-axes provide a means of describing each point on the plane 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 was

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,

(x, y)
(-3, 9)
(-1.5, 2.25)
(-1, 1)
(-0.5, 0.25)
(0, 0)
(0.5, 0.25)
(1, 1)
(1.5, 2.25)
(3, 9)

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.

¹ The image associated with this article illustrates the parabola y = x².