Is there a recognized category of functions that have the following features:
So, y=-ax^2, y = -a(abs)(x), y = A/(1+x^2) would be members of the category.
Any odd function, even functions such as -x^4 + x^2, or repeating functions such as y=sin(x) wouldn’t.
I haven’t heard of a name for them, but for cases where there is such a peak at one point, and which asymptotically approach zero for x going to infinity, I have privately called them “bump” functions. They are nonzero only in a restricted location, and there are only a few physical or mathematical reasons for their functional form, so if you see one you can tell what’s responsible for them.
Probably the most common form is the Gaussian, or normal distribution function, but there’s also the Lorentzian, the Sinc function, the Inverse Hyperbolic Cosine, the Binomial Distribution, the Poisson Distribution, the Sinc Squared function, the “Jinc” function, and so on.
Unimodal functions.
That looks perfectly crommulent! Thank you!
I don’t see in the definition where y asymptotically approaches zero, unless by “smaller” he means “smaller magnitude” as opposed to “lesser value.”
Regardng Post #2, I do not believe that Schwartz functions or bump functions are required to be unimodal, as opposed to (in the “bump” case) simply being smooth and having compact support.
It’s not in his definition. I was simply referring to my definition of a related case.
I was poking around Cal’s curves. First of all,a shout out to the Witch of Agnesi which is the bitchingest name for a curve I have ever seen.
At any rate, Wolfram Alpha gives an equation of the Lorentzian function as follows: 1/(1+x^2). Elsewhere it gives the equation as being much more complicated, involving pi and an upside down L.
Is an equation in the form (C - bx)/(1 + bx + ax^2) a Lorentzian function?
Your link says the Lorentzian function is essentially C/(x² + a²) or C / ( (x - x₀)² + a² ). So C / (ax² + bx + 1) is of this form, but not with the second term you have added.
I would say “function with a single local maximum”. But I don’t know if there’s a shorter name than that.
Unimodal was the name I encountered for such functions in the past while studying chaotic dynamics. Unimodality - Wikipedia Sample search I did to verify this: unimodal maps chaos - Google Search
That might work … provided that in y = x - x^3, (-∞) is considered a “local” point. (If so, I think the shorter name you’re looking for is … unimodal. 
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Quasi-concave functions can be useful, though that is more general than what the OP describes. That basically means that if you pick two points x and y, and p is between x and y, then f§ ≥ min { f(x), f(y) }.
That’s what I was going to say. The OP’s description immediately brings the classic bell curve to mind, for me anyway.
OP, here. Yes, a quasi Gaussian distribution is one of the functions I had in mind, but I needed a term that would also include an inverted parabola and an inverted V. I have three models, one in each of those shapes, so I was looking for a general term for all of them, and unimodal was just the thing. My quasi-Gaussian distribution failed a test of normality, though, and that’s why I had to go with the form C/1+ax+bx^2.
At any rate, I’m enjoying the discussion! As always when I post my naive math questions, I not only get the answer I’m looking for, but learn so much more.