Points of Inflection

... in a relationship?

Yes. Relationships (especially committed ones) are like mathematical functions. This is my thesis. See if you agree.

(Stay with me now, this may get obscure.)

I've been doing a little math tutoring at the college level the last few weeks, plus D was ooohing this week over her new graphing calculator for college algebra, so when the conversation over dinner drifted into polynomials of various degrees it wasn't too surprising. Geeky, maybe, but not surprising. It's always been that kind of relationship. :P

We then went over the relation between the degree of a polynomial function and the number of bends the curve of it makes on a graph. The relationship is:

Highest exponent = Degree of polynomial. The curve has (Degree - 1) points of inflection.

The point of inflection is where the curve bends and heads in another direction (related to the first, yes, still connected to the curve, yes, but different in direction and shape.)

A straight line is of the form Y = aX + b where a is the slope and b the Y-intercept (which is the value of X at which the line crosses the Y axis), hence what's known as the "slope-intercept" form. Slope is rise/run, or how many units of Y you get for every additional X unit. So Y = 2X + 3 looks like this:



Okay, so with that as background, every time you multiply that function by an additional X term, you get a different looking graph. Suppose you multiply the original function by another X. Then you get Y = 2X^2 + 3X (the X^2 is pronounced x-squared), which looks like:



Oooh. It bends! Now it's no longer a line, but a curve. And the place where it bends is the point of inflection, the point where the line changes direction.

Now, if you do that again, this time, say, by multiplying 2X^2 + 3X times (x - 1), and then do something else to it, too (like subtract 1), you get this: Y = 2X^3 + x^2 - 3x - 1, which looks like:




















Hey! It bends twice now. Funky looking thing. Notice that the highest exponent is 3 (from the first term which contains X to the 3rd power, or X-cubed.) So the degree of the function is 3, and it bends (3 - 1) or 2 times.

So that's the deal. The higher the order of the function, the greater the degree of the polynomial, the more points of inflection it has and the more times it changes direction.

***********

So? So what's my point?

We're clearing the supper dishes, and I'm finishing up this little discourse and D says. "Points of inflection. You mean like we have in our relationship." And I laugh. Ha. Hahaha. But then...

Bing! (the lightbulb goes on...)

Hey. I think you have something there.

Yeah. Yeah, yeah, yeah.

I think it's true. Relationships are like that. They can get more complex over time (or not. Some stay simple like straight lines.)

But I think there are often major life events that can function like a new factor that affects the curve of your relationship, like multiplying by (X-1) or something. And then, as you recalculate...

your relationship takes a turn. It becomes a higher-order equation, a function of a greater degree of complexity. It's related to the one before it, but... it's changed into something more complex, less straightforward, harder to describe.

And I suppose it happens the other way, too - you can divide by a "factor", and ... move to something simpler, easier to deal with and display. Different life events affect a relationship in different ways.

And this last year had a few life events for me. They affected my relationships. Some got more complex, and some got simpler. Some grew in degree, some shrank. Some just shifted left or right, up or down on the graph, as if you just multiplied by or added a number, but not another X, another variable.

How can this happen in a relationship? A change point, a tipping point, a point at which the relationship changes to one of another degree? Obviously a loss of job, losing a loved one, making a career change, going back to school, leaving home or moving back, or just saying goodbye...

all those things can affect a relationship.

But so can little things which may bring profound insights. Here's an example of something simple - but profound - for this couple. It's 8 minutes, but worth watching. (especially if you lasted this long!) ;)

Anyway, what do you think? Do relationships have points of inflection? Points at which they curve and go in a new direction? Changes in their level of complexity (greater or lesser)?

The X's and Y's make me think of men and women mostly, and how relationships are like functions. You do something to X... and there's a resulting change in Y. Or vice versa. When one person changes, the other one feels it... and adapts.

Makes sense to me. But then, I do math for a living. And occasionally - just for grins. :)