Previous page : y^3+y^2=(x^4)y+x^2

Let us look at

**Zeroes**

Let us start with possible axes intercepts. If *y*=0 we get *x*^{2}=0, so *x*=0. Multiplying all by *x* we get

So at *x*=0 we would get *y*=0, but the relation is not defined at that point, since we then would divide by 0, but we could expect the graph to exist in a neighbourhood of that point.

**Critical point**

If we now differentiate the second equation with respect to *x* we get

For *y*´=0 we get

Or

We would thus expect to find the slope to be zero somewhere along that line.

Using the second equation to eliminate *y* in the first we get

or

This gives us

That, obviously, has one solution at *x*=0. Dividing through by *x*^{2} and 8 gives us

or

(I had to grab my calculator there.) Substituting that back into

gives us

Making *y*´ the subject in

This will be undefined, i.e. *x*´=0 when

or

We now substitute that into

to get

or

This gives us

or

And, for that value of *y*,

**Asymptotes**

If x goes toward infinity then

Why so? If x goes toward infinity and say y would go toward anything but infinity, then y/x would go toward 0, but at the same time would the left-hand side go toward infinity, and that will not work. If we instead assume y goes toward infinity, then *y*^{3} would have to go toward infinity in about the same phase as *x*^{2}, and therefore *x* would go way faster toward infinity than *y*, so *y*/*x* would go toward 0. We will thus (probably) have

As an asymptote, and that would be true in both the positive and the negative x direction, as is not too hard to argue for.

Also, if x goes toward 0, then we could either have that *y* would go to 0 too (but faster) so that both sides go toward 0 (as we already saw is plausible from the argument about the zeroes in the beginning). Alternatively, *y* might go toward infinity so that

Goes toward

or

As often is, it is rather hard to find the zeroes of the second derivative, so I will not even try to find the points of intersection. Anyhow, putting these facts together we get:

The solid lines are the asymptotes, and the dark lines are points with known slopes.

Before looking at the actual graph, try to figure out what it looks like.

Anyhow, here it is:

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The green line is our graph.

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