The general graph of an Odd Degree Polynomial

The text uses the following graph, or one quite similar to it, as the graph of a general odd degree polynomial with a positive leading coefficient.

That particular graph was generated with a window setting of

Unfortunately, the graph of the polynomial gives a number of erroneous impressions. The only thing that we do know for sure is that the graph must go down to the left and up to the right. We do not know what happens in the middle. The image shown above implies that the graph crosses the x-axis in three places. This is not necessarily so. It is true that the graph of an odd degree polynomial must cross the x-axis in at least one place. But a general odd degree polynomial may cross in many places, any number of which may be to the left or to the right of the origin. We just do not know. The most we can say about the graph of an odd degree polynomial with a leading positive coefficient is that it will appear as

where we do not know what happens inside the middle box and we do not know where the box is with respect to the coordinate axes. For example, each of the following is a graph of an odd degree polynomial with a leading positive coefficient.

On the other hand, the text presents the graph of a general odd degree polynomial with a negative leading coefficient as

Again, this graph may give some wrong impressions. All that we know for sure is that the graph must go up at the left and down at the right. The general graph should be presented as

where, again, we do not know what happens in side the middle box and we do not know where that box is in relation to the coordinate axes. For example, each of the following is a graph of an odd degree polynomial with a leading negative coefficient.