Talk:Parent function

This article

I created this article based on my query on the Wikipedia:Reference desk (2013-03-25). It's only a stub, and some more discussion about it (below) needs to ensue. — Loadmaster (talk) 17:32, 25 March 2013 (UTC)

Ambiguous definition

The definition seems to be ambiguous for cubics; x^3-x has two local extrema which is not true of x^3. Robinh (talk) 04:30, 25 March 2013 (UTC)

From what I can tell from a Google search for "parent function", this term is used in various textbooks with no fixed meaning. It generally refers to one of a collection of basic functions whose graphs are good for students to memorize, and from which the graphs of other related functions can be found by applying graph transformations. But there is no consensus on exactly which functions make up the set of "parent functions". A PDF from Glencoe lists only y = x and y = x². Mathwords.com includes y = x, y = x², y = x³, y = x^1/2, y = x^1/3, y = |x|, y = 1/x, y = a^x, y = log_a x, and the trig functions and inverse trig functions. About.com does not explicitly name a list of parent functions, but describes several classes of algebraic functions, excluding polynomials of degree higher than 2 and all radicals. A PDF from faculty.gg.uwyo.edu even includes $f(x)={\frac {(x^{2}+1)(x-2)}{(x+1)(x-2)}}$ as a parent function! Since there is no consensus on what constitutes a "parent function," and none of these sources give anything near a rigorous definition, I don't think that "parent function" is a meaningful mathematical concept for a Wikipedia article. It's just a vague concept some textbook authors have invented. —Bkell (talk) 04:44, 25 March 2013 (UTC)

I can see the definition as "the simplification of a family of related formulas" as useful, but obviously vague. Simple cases like x for all linear formulas (ax + b) and x² for all quadratics (ax² + bx + c) are obvious. But more complex polynomials and functions with non-integral exponents make the idea of "simple, but not too simple" difficult to state a rigorous definition. An example someone else raised: is the parent function for 2x³ + 4x² + 1 something like x³ or something that "looks" more like it, such as x³ + x² ? — Loadmaster (talk) 17:42, 25 March 2013 (UTC)

Note that Mathwords.com says that the parent function of xⁿ is x² whenever n is even, and x³ whenever n is odd. The reason they say this is apparently that these graphs have "roughly" the same shape, and the same end behavior. Clearly there is no rigorous definition here. —Bkell (talk) 21:08, 25 March 2013 (UTC)

I haven't come across the term "parent function" before, but the concept is familiar. I've extended the article a bit, allowing different interpretations for higher polynomials and explaining the motivation for the concept. Please improve on my attempt. Dbfirs 09:51, 26 March 2013 (UTC)

Good job. I've cleaned up some of the formulas. What we need are more cites and references, if good ones exist. Also, a diagram or two would be very nice to have. — Loadmaster (talk) 18:23, 26 March 2013 (UTC)

Thanks. I'm never sure when to use "< math>" and when not. If we can't find convincing references for "Parent function", then we could rename the article (perhaps "Graph sketching" which is currently a poor redirect) and redirect to it. Alternatively, we could add the content as a major section in curve sketching which gives only general methods and misses out the parent function method. Dbfirs 21:04, 26 March 2013 (UTC)

I agree, seems very dubious and ill defined to me. What means simplest representative? Is sin simpler than cos? This shows that this is just a matter of definition, i.e., name giving. It is not a mathematical concept. That should be made clear at the beginning of the article. — MFH:Talk 01:05, 10 November 2020 (UTC)