Does a normal curve ever intersect the x-axis? Why or why not?

No.

The x-axis is an asymptote for the normal curve. This means that each of the two tails of the curve approaches the x-axis at a thinner and thinner angle but never quite reaches it.

Additional Discussion:

Theoretically, these extremes of the normal curve extend infinitely in either direction, given their asymptotic relation to the x-axis. However, many phenomena that are considered to be reasonably well-approximated by a normal curve actually have finite limits. Heights, weights, test scores, and IQs are examples. There are lower limits of zero in each of these cases (except on tests like the SAT). Test scores also have upper limits.

(Realistically, heights and weights are also limited on their upper end, though not by any SPECIFIC quantity. Realistically, the same is true on the lower end. Zero is not a realistic height or weight -- at least not for a human! But there is no specific lowest possible height or weight.)

In such cases, when the tails of the curve are not genuinely extending infinitely, the normal curve is often drawn as though it "rests on" or "meets" the x-axis but does not quite intersect it. The asymptotic relationship is maintained.

In the case of test scores, it makes some degree of intuitive sense that the probability never quite reaches zero in the tails of the distribution. There is SOME probability, however small, of there being one or more "0" scores on most tests. The same is true of perfect "100" scores.

On the other hand, however, one might argue that there is NO chance of a "-1" score on a test, nor is there any chance of a score greater than "100," in many cases. You may be able to instigate an interesting conversation with your teacher/professor, if you ask about types of data like this, data that seem to many to be well-represented by a normal curve but that also seem to have clear examples of values for which the probability SHOULD be zero (i.e. ON the x-axis).

Why are these types of data considered good examples of applications of the normal curve?

In any case, it seems that the notion of a "normal curve" most faithfully applies to types of data that do NOT have upper and lower bounds on their ranges of possible values. For data like this, the normal curve tells us that no matter how small or great the score considered, there is at least some minute probability of that score occurring. Hence the asymptotic relation of the tails to the x-axis.