File talk:Cubic graph special points.svg

This is a nice graph, but I'd like to request a couple of changes to it:

The vertical and horizontal axes are scaled very differently. So for example the distance from (0, 0) to (0, 432) is nowhere near being 432 times the distance from (0, 0) to (1, 0); and the straight line showing f "(x), which algebraically has a slope of 6, visually appears with a slope far smaller than 1. Could you find an example with some more convenient numbers so the scaling would be equal? If not, I request that you add this sentence to the end of the caption:

The vertical scale is compressed relative to the horizontal scale for ease of viewing.

In purple at the bottom of the left and right parts of the graph, it says

f(x) curve concave downwards

and

f(x) curve concave upwards

In formal math terminology, however, "concave downwards" is referred to as "concave", and "concave upwards" as "convex".

I appreciate your work on this and hope you can make these improvements! Duoduoduo (talk) 03:16, 19 December 2011 (UTC)[reply]

Excellent, but it can be slightly improved,

Error: “minus” missing at turning point …(8,– 400) instead … (8,400).

Dash-dot style is usual for symmetry lines – perhaps you could mark turning points ordinates by dash-dash one.

Y vs. X drawing ratio should be quoted. Is it now 54:1? All captions can be roomed, I hope, if you would compress it to 16:1.

Suggestion: a line of tangency at inflection (and uneven symmetry point) is of utmost importance since its slope entirely links a graph with the article on the following way:

y = x^3 – 3x^2 – 144x + 432 = (x – 1)^3 – 147(x – 1) + 286 where all of three variables are the inflection point properties:

1 = –b/(3a) = s is its abscise (“s” for Symmetry and for Shift of Y-axis towards inflection point),

286 = (2b^3 – 9abc + 27a^2d)/(27a^2) = q is its ordinate and

–147 = (3ac – b^2)/3a^2 = p < 0 its slope.

Variables s, q and p are forming

y = (x – s)^3 +p(x – s) + q which is depressed monic cubic for x – s (see few chapters below 3.3 Reduction to monic trinomial).

Y = –147(x – 1) +286 = –147x +433 is analytical form of this tangency line passing through the points (1,286) and (–3,874) so it can be easily constructed in order this image to be completed.

Regards

188.127.120.236 (talk) 16:37, 19 December 2011 (UTC)Stap — Preceding unsigned comment added by 188.127.120.236 (talk) 14:59, 19 December 2011 (UTC)[reply]

Thanks for your feedback, Duoduoduo and user at 188.127.120.236. I've updated the following:

Turning point y-coordinate.
Concave/convex labels (though the original was also valid, and is the form used in Inflection point).
Refine note about x/y scales.

but have kept the following:

Dot-dash lines are normally indicate axes of symmetry in engineering drawings, but in graphs, they need not mean this. The line style was chosen to distinguish from the solid, dotted and dashed ones for the functions.
The main reason for choosing this function was that it is (as far as I know), the simplest one with distinct special points with non-zero integer coordinates. It keeps the numbers clearer for readers, and SVG is not ideal for typesetting square roots. I had already added "Note that the x and y axes have different scales" in the image description. I don't understand what you mean by "roomed".

I'll think about how to include the tangent without it being too cluttered. cmɢʟee ☎✉ 12:48, 3 August 2012 (UTC)[reply]

I've added a tangent at the inflection point, but can't see why the above calculation is so convoluted. Isn't it simply a line of gradient f'(X) and passing through (X, f(X)), i.e. y = x f(X) + (f(0) - X f(X)), where X satisfies f"(X) = 0? cmɢʟee ☎✉ 19:35, 3 August 2012 (UTC)[reply]