When the story tasks you with hunting down the elder dragons and following blue tracks, you will also start finding purple tracks on your hunts. Investigating these purple tracks are for tempered monsters. Investigating a tempered track gives you a chance of unlocking a new investigation. You may not get a new investigation on your first try, but it's a fairly large chance to get one.
Go to the researchers where you accept your investigations and you will find a new kind of investigation with a purple aura around the tempered monster. These monsters have a good deal more fight than your normal hunts. Each investigation can be done multiple times. The number of tries given is random for each investigation.
There are normal, capture, and great hunts that have tempered monsters. My first tempered investigation was a great hunt, but only one monster was tempered. There may be some luck involved in getting good investigations for the 50 kills. Just keep collecting purple tracks for more investigations.
There are tempered quests that halt your hunter rank after the story. At rank 29 you must fight a tempered Bazelgeuse twice and rank 49 is a tempered Kirin. These are repeatable.
I would not recommend taking on a tempered Kirin with a group of randoms. Either have people you know can help, or do it solo. It seems crazy, but I completed the quest my first try solo with a bow. Just don't start a long combo, because Kirin attacks very fast.