Extra Credit

What is the role of extra credit? There are students in my class who were F's at the end of the semester. I offered an extra credit project. Those who did it were able to finish with a D in the class. But
a) I felt like I am giving them a cop-out from doing work during the rest of the semester, because by doing the extra credit assignment, they didn't have to make up half of the other assignments that they had missing.
b) Although my target audience with the extra credit were the D's and F's of the class, the majority of the A students completed it as well, causing a false inflation of the grades on that end.
c) The content and skills required for the extra credit is difficult to adjust. Since I'm targeting the lower-level students, it should be "fun", and a good chance to be creative (to provide differentiation by interest), as well as appropriately scaffolded. At the same time, it should still challenge students to learn content and apply skills.

The criteria for extra credit should be:
A) Not offered in a way that detracts from classwork completion
----- Differentiated and scaffolded
----- of low point values
----- Perhaps available only if you have x assignments missing?
B) Not artificially inflate grades, while still allowing students to earn points
C) A learning opportunity for all students who attempt it


The question then becomes, why have it as extra credit? Shouldn't I be able to differentiate within the classroom?