I recently played in a rapid arena and I (1250 ish) beat a 1780 (Santiago2000) and he lost 52 rating points where I only gained 15. I figured it was probably a new account so I checked but he has over 900 games of rapid. His match history seems to show some large elo jumps game to game, much greater than the +/- 8 I receive. Is there a way to change the number of points received/lost per game? If not how is he gaining and losing so much between games?

No. The site uses the Glicko rating system that includes a measure of rating uncertainty called the rating deviation (RD) value. Players with a higher RD values will get larger rating changes after games.

Players with few games in a rating pool and accounts with longer gaps in rated games in a pool, regardless of the total previously played, will have higher RD values. Your opponent must have had a higher value

How do ratings work on Chess.com?

Everyone on Chess.com has a rating. The purpose of this rating is to try to show the level of skill of each player, to better match you against other players of equal rating. 

You’ve probably noticed that after you win or lose a game, your rating doesn’t always go up or down by the same amount. What causes this difference? 

A few things can affect how much your rating changes after you play a game. Some things that can affect this are:

1. The difference in rating between you and your opponent
2. How confident we are of your rating
3. How confident we are of your opponent’s rating

Number one is pretty straight forward. If you win against someone who has a much higher rating than you, your rating will go up by a higher amount than if you win against someone of your same rating, or of a lower rating. 

Similarly, if you lose against someone with a lower rating, your rating will go down more than if you lost to someone with a higher rating. 

Numbers two and three on the list are a bit more complicated. It might sound strange, but confidence in the rating is the main reason for big changes in your rating after a game. 

Chess.com uses the Glicko rating system, and part of this system is a number called a ‘rating deviation’ or RD, which measures how sure we are of what your rating is. If you've never played any games on Chess.com before, the system has no idea of what your 'real' rating is, so it will move your rating up and down a lot during the first few games you play.

If you aren't new, but haven't played any games for a while, your RD will also go up, and the next game you play might change your rating by a lot. 

This is because Chess.com hasn’t seen any games from you in a while, and there is no way to tell whether you’ve been practicing and getting better other places, or if you haven’t been playing at all. 

The same applies to your opponents as well. If you lose to someone who is several hundred rating points lower than you, but with a very high RD (they haven't played in a long time), your rating won’t change much, because we can’t be sure that is their actual rating.

If you’re curious about more of the inner workings of the rating system, check the articles here and here! 

Our advice to you, though, is to just play games and have fun, and let the rating sort itself out!

If you read the above, you can take a look at Santiago2000's rapid rating and you will see that he had only played 15 games this year. Therefore, it seems likely that the RD for his rapid rating was much higher than yours.

Santiago2000's RD for rapid is currently 90 while yours is half that.

Yours is 44:

Slightly related: In the past, I'm certain there was a "rating adjustment" note in the right column -- I used to click it occasionally to see what I would have lost if I lost, or drawn, and so on. But now it's not there anymore.

This Help article - https://support.chess.com/article/1216-how-many-points-do-i-get-for-winning-how-do-i-know - says it's still there, but I'm not seeing it on any of my games. For example, I'm curious how many points I would have lost for this game: https://www.chess.com/game/live/97189875363 (accidentally played a rated game versus my son). I'm sure I would have lost a bundle of points if he'd beaten me, but the "rating adjustment" item is gone now.