Picture from the other side to show right side burnout.

From the tiny glimpse of your house, I imagine that it's quite nice looking. The tree is not only showing hostility toward your grass, but it's obscuring a large portion of the view of your house ... and I'm presume the view from your porch into the neighborhood, if you want that. (I don't know.) Trees do not normally maintain themselves relative to man's needs. Man must shape the tree according to his wishes. I wish you would limb this tree up, get more light under it (or you will ONLY be able to grow ivy!) and create a better view of your house. If you are using the tree canopy for privacy, well, then there's not much that can be hoped for.

Thanks for the advice Yardvaark. Limbing up the tree is actually something I was thinking about doing. It definitely does block the view of my house. I'm not so concerned about privacy as much as I am just butchering it. I actually have even thought of removing it altogether but it does provide nice shade from the summer sun. I also live in a neighborhood that was established in the thirties so there is approximately one huge tree in every yard. And I don't want to be the neighbor who moves in and takes one down.

The degree of slope looks like turf would be difficult to keep mowed as well as watered, and ripe for conversion to a garden setting of plantings that will accept the different sun/shade conditions. The tree looks as if it already does have a fairly open canopy for the lower portion of the tree, not knowing the type of tree it may be premature to recommend heavy pruning. New plantings should probably reflect the varied sun/shade aspects.

Ivy isn't the only shade ground cover, and ornamental grasses or sedges can do better in both sun and shade conditions with less irrigation than mown lawn. Look around your neighborhood to get ideas on what plants do well in similar circumstances for ideas.

If you post some good pictures of the tree taken from different sides, you could probably get help in determining what branches should be removed. You might need to remove some and post more pictures when more of the tree can be seen. Also, take a picture from across the street that shows the overall relationship the tree has to the house.

You might speak to how much maintenance you're interested in doing on a regular basis. Are you interested in expanding you gardening interests, or keeping the yard simple with a better landscape presentation and low maintenance? Converting the slopes to groundcover would be lower maintenance than mowing once the plantings filled in sufficiently to keep weed infiltration down.

Couple of thoughts ...
1. Grass- I am not personally much of a velvet green, get-the-tweezers-out sort of lawn person, but the grass grows pretty well in my yard even though we only sort of haphazardly do the things you're supposed to do with grass.
But, one thing I know is pretty standard in my zone 6 is that fall seeding and watering in of the lawn is best. Spring seeding is second best. Fall seeding lets the roots get started and gives greater strength to the grass to face the summer heat. Zone 6 for me means cold winters, inadequate snow cover to think of it as insulating plants, strong winds, big spring storms, and hot humid summers.

2. Water and Grass type- My front yard also has a large "neighborhood" tree that is, I guess, about 40 years old, not really proportional to the house, and great at lowering our AC bills. Grass grows under it pretty well. But ... we do water it and our yard is only slightly sloped. But, because of the type of grass ... oh, don't make me think of the type; it's been a while since I knew that ... Well, zone 6 is sort of borderline between the summer grass and winter grass or whatever. Anyway, you can use blends here, but many yards brown out in August. Not dead, just brown. When the cooler fall weather comes, like now, it all greens up again.
Do other yards with big trees in your neighborhood have grass?

3. Ground covers- Like others, I'm wondering what sort of tree you have. It still stuns me how wide ranging tree roots can be. Our big front yard tree really has roots throughout the lawn. It may mean that my lawn days are numbered although all is presently okay. And, my tree is a maple, one of the types notorious for its surface roots.
The thing about ground covers is that you'd still need to know not only what sort of tree is competing for the ground, but also what sort of soil the ground covers will be living in. Slope means water drains quickly. Slope that's hard-packed clay means very little water can be taken in for immature plants, whether grass, which, by the way, is a ground cover, or by some other ground cover. These plants will do fine in spring when clay soil may still hold winter moisture. But let the sun bake it hard and young plants will soon be fried.

Random thoughts. Whatever you do, knowing the plants, knowing the condition of the soil, will help you decide.