I would agree that it might be wise to have her screened for depression and realize that it might be a bit of culture shock for her to be living with you guys and have different expectations made of her. Talk to her about anything that she finds the least bit interesting - sometimes it takes a while for them to discover something they can be passionate about.
My stepsons (16 and 17) are good kids, but seem to suffer from the same problem - or at least until recently. They live with their mom and she's a nice person and a loving mom but hasn't always been much of pusher when it seemed to me they really needed a push. If they ever took an interest in anything or started an activity, then decided they didn't want to do it anymore, she let them quit before they really gave it more of a chance (this includes cross country, football, tennis, guitar and drum lessons). So now neither of them is really involved in anything beyond video games and skateboarding, and their grades are not always what they could be. Then earlier this year the 17 yo discovered photography and now that he is a senior in high school, he is hoping to get into a particular college to pursue it further. He is also now working part-time at a fast food burger chain. The 16 yo is in 10th grade and is still being a bit of a lump but hopefully he'll figure something out and get it together soon. It is hard when they don't live with us and I and his dad have limited influence - whenever Dad has tried to talk to Mom about this stuff, she gets fairly defensive and not much gets accomplished (and neither she nor he went to college, whereas I did, for what it's worth - something I think that can affect one's perspective on how important a college education can be).
If your community offers classes through parks and rec, maybe you could sign the both of you up for something together. Let her see what is available and see if anything peaks her interest - cooking, photography, yoga, whatever. Let it be a chance for the both of you to doing something fun together and hopefully that will get her feeling more positive about herself as well as you. You could also consider volunteer work and then maybe she will see that she really doesn't have it that bad.
You did not mention her age or what grade she is in, but maybe she needs to think about going to a vocational/technical high school instead of a regular one - she can still go to college after graduation but it might give her taste of what the alternatives are if college is not an option. She can also think about getting a part-time job like my stepson did - at least it will give her something to do with her time and start teaching her some work ethics and lessons in real life. It can also help her to see that her career options might be limited if she does not go to college. If she is struggling with her grades, it could be a learning issue and you might need to get more involved with her teachers - then maybe she will see that you care enough to show up and she'll start to care too.
Finally, you might just have to lay it all on the line - if she can't get better grades, she will not get accepted to the college of her choice. Unfortunately, some people (teens included) can only learn the hard way - they have to screw up and make poor choices and realize that they need to do things differently if they want to turn their life around.
Please try to be as upbeat and supportive and positive as possible - if you are giving off negative vibes, thinking of her as "bratty" and "unmotivated", she can probably sense that and that will just cause her to withdraw more. She might think that she will fail no matter what, so what's the point in trying. Let her know that she is smart, she has a good head on her shoulders, and all she has to do is put in the effort - she has the brains, but the rest is up to her. It's not her fault that her mother is the way she is and she had the upbringing she's had up to this point.
Of course, I have to ask: where is her dad in all this?