Plot a curve having no equation

Question

I don't have the equation of the curve. The curve has a maxima and a minima as the following fig. How to plot such smooth graph using tex. Please suggest. Is it possible to suggest an equation of the curve have the same structure? How to get y such that y passing through (8,0) and (0,5) has a mimima at suppose (2, 4) and maxima at (6, 8)

Edit:

Curve of y=-(x+1)(x-2)(x-3)

Answer 1

Of course to plot something it must ... be something. That is to say, in this case, it must be some function.

But given the constraints, you can pick anything you want, as long as it meets the constraints. Judging from the picture, the main constraints that I see represented are:

• It goes to infinity as $x\to-\infty$ and it goes to $-\infty$ as $x\to\infty$.
• It has a local minimum and maximum.
• It is "smooth" (continuous derivatives of all orders, let's say).

So $-x(x-1)(x+1)$ should suffice, which you can easily plot in almost any mathematical plotting software.

If the graph is supposed to fit in some kind of a box with the origin in the lower-left corner as it seems it might be in the picture, then you can shift and scale the graph until it fits. To scale it, use

$$ -rx(x-1)(x+1) $$

for any $r$ you like, to shrink or expand it vertically. To shrink or expand horizontally

$$-r(sx)(sx-1)(sx+1)$$

To translate this up and right by the vector $\langle h,k\rangle$, use

$$-r(s[x-h])(s[x-h]-1)(s[x-h]+1) + k$$

So you can play with the parameters $r,s,h,k$ until it fits where you want it.