Why do you get side band frequencies for amplitude modulation (AM)?

I thought that the frequency was constant for AM modulation, and just the amplitude was modulated. So why are there a range of frequency's (side bands around the baseband) when the signal is plotted on a frequency domain graph?
Thanks

The frequency for AM modulation is constant, but we have 3 signals that compose the AM singal: the carrier, the upper side band, and the lower side band of the envelope.

Yes the carrier is constant, the amplitude isn't, so look at it this way which is simple.

A*sin(w1t) is just the carrier with constant frequency and constant amplitude with no information => w1 is some frequency to be different that the modulation frequency.

But what if I replace the, "A," constant with a variable function, or information, like say, ( 1 + K*sin(wt)) , we get the carrier sin(w1tt) +/- the sideband frequency.

So if you can understand the trig function for products, then that's really the reason.

the carrier is constant sidebands result from the carrier plus the modulation frequency....upper sideband and the carrier minus the modulation frequency.....lower sideband. there will also be 2nd and 3rd harmonics approx 30 db lower.

pure sine signal has theoretical 0Hz bandwidht, any modulated signal deviates from pure sine and thus has wider bandwidht. the information modulated onto carrier has to "fit" somewhere on the em spectrum, the data signal has its own bandwidht, without compression algorythms or loss of data you cant fit it onto less, frequency shifting it to RF frequency does not reduce its bandwidht