Variation Analysis, Parabola

central equation:      y² = 4cx

The question of interest is, how does the change in c, the distance to the focus, affect the shape of the parabola?

First, for a given c, as |x|↑, y²↑. And as |x|↓, y²↓. One look at the graph of a parabola will convince you of this.

Now, allow c to increase for a fixed x. The equation dictates that y must increase as well .

(y² ↑ = 4c↑ x) The graph below shows 5 parabolas with different values for c.

The smallest c is 1/16 (4c = 1/4, so c = 1/16). This is the magenta parabola which is almost flat along the X⁺ axis whose focal point c is very close to the origin (x=1/16).

As c decreases toward zero the parabola comes closer and closer to the X⁺ axis. Y still increases as X increases, but for very small c the change is slight.

c can become as close to zero as we want, the parabola still does not touch the X⁺ axis. If we set c to 0 then the parabola collapses onto the X⁺ axis. Substitute 0 into the central equation and you will find y² = 0 (for all x), but, we no longer have a parabola, we have a horizontal line y = 0.

The largest value of c shown is 8 (4c = 32, so c = 8). This is the cyan parabola. It opens close to the Y axis. The focus of this parabola is at x = 8. As we increase c the parabola will flatten along the Y axis, but it will never touch it. In this case, as c increases, 2, 4, etc., there's always a number larger than the last number in this sequence. We say this sequence has no bound, there is no maximum number here. As c becomes very large for a given fixed x, the y values of these parabolas continue to get larger. y²↑= 4(c↑)x

And for a given fixed y, as c continues to get larger the x values tend toward 0.

y² = 4(c↑)|x|↓

In this case there is no limiting value for c, so the parabola opens toward the Y⁺ axis (the cyan parabola) but will never touch it (except at x = 0). At x = 0 we have a vertical line x=0, that is, the Y axis.