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 = ax^{2} + bx + c

If a = 0, the term ax

^{2}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 = x

^{2}. 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)

## 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.**References:**

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