Square Root Curve (capped) vs. Bell Curve

Presently trying to understand the 2 curves above.

Why a teacher would use a square root curve then put a 20 pt cap on the curve is confusing. If you have to cap it, why use it at all? The curve is designed to produce certain objective results. One article said this was an absolute curve and each student received pts independent of what classmates scored. It was designed to bunch the scores (bring them closer together).

11*SQRT of raw score is what teacher used to curve exam that had disastrous results (many, many low scores).

However, she capped the curve at 20 pts because she did not want anyone getting more than 20 pts. This was difficult for me to understand because it looked like she was manipulating the outcome with no stated objective.

The people who scored above a certain score (let's say 76) received the full benefit of the curve. They got all of their curve points. However, the people who scored less than the 76 did not receive the full benefit of the curve. If the raw score was supposed to be increased by 30 pts, she only gave the student 20 points.

How do you look at this objectively and convince a student (after the fact) that this was a fair way to curve?

Also, I am trying to understand why one school system in particular has disallowed the bell curve. It is stated in the policy that this curve may not be used. There are no explanations, just the statement. I know with the bell curve, a student's curved score depends on how well the other classmates perform. Perhaps this is why the school rejects this one. But how does putting a cap on a square root curve unannounced, unplanned make it a better curve?

More Background:

The teacher never discussed the curve because she felt like she had the right to determine how the final grade was determined. However, she never told the students if and how she would curve raw scores prior to testing.

She has never been willing to state an objective for curving.

She uses College Board AP Exams almost exclusively (the difficulty level is extreme for a classroom exam)

On a given test day it is possible for the same class to have different tests. Only one student received a college board problem that most of the College Board Exam Candidates could not answer. After analyzing one of the classroom tests, I discovered that (this one problem given to one student) only 12% of College Board Exam Candidates answered it correctly.

Help with understanding the curve issue is appreciated.