I am an adherent of the school that teaches that the embouchure remain unchanged throughout the normal playing range of the saxophone (low Bb to high F). However, the speed and direction of the airstream as well as the shape inside the mouth does change.
Think about what has to take place when playing rapid arpeggios and octave leaps if one loosens the embouchure for lower notes and then tightens the embouchure for high notes. This is working entirely too hard, the sax doesn't require it, and the intonation suffers because of it.
The embouchure "test" on alto that I use is to play the neck and mouthpiece alone and produce the pitch Ab concert. The pitch on tenor is E concert. That is the embouchure tightness setting that will work best throughout the normal range of the saxophone. On alto, for example that corresponds to the note F on the top line of the staff. When the sax is assembled and the note F is played with the embouchure setting that produced F on the mouthpiece and neck, the note just sings.
Keeping the embouchure the same, with a greater volume (not speed) of air, low Bb should "sing" as well if the saxophone is leak free. With that same embouchure and a faster airstream high F above the staff should "sing" as well. High F may also require raising the back of the tongue just a bit as well.
Some "jazzers" use only "subtone" for their low notes because of the quality of sound it produces. I do that as well when playing in that idiom, but I also believe it is important to be able to play the low register with a "legit" sound as well. Some folks confuse the dropping the jaw motion to produce subtone with having to "relax" the embouchure to play the low notes. They are really two different things. Playing the lowest notes on the saxophone does not require relaxing the embouchure. Playing the lowest notes with the breathy subtone sound does require an embouchure change to produce that sound quality.
Just a word about the altissimo register. There are some players who claim to be able to play the altissimo without tightening the embouchure at all. I am not that good a player, nor have I spent considerable time practicing in that area. The best analogy I know for how to produce harmonics on a given fingering and to play altissimo notes is "whistling". To go higher when whistling, one raises the back of the tongue which increases the speed of the air and decreases the volume inside the mouth. To go lower when whistling, one lowers the back of the tongue which slows the air and increases the volume inside the mouth.