Answer: I think you are probably growing a variegated Dianella tasmanica. It can grow to four feet tall and wide (when planted in the ground) and with its variegation actually prefers even more shade than most kinds of Dianella. I would try growing it in at most some gentle morning sun or dappled light all day -- a shaded yet bright location with indirect sun would be best outside.
This big plant should be a fairly vigorous grower spreading by rhizomes. I would suggest repotting it into a wider container using a well drained soil mix as I think it may be rootbound. (Maybe wait until late winter to do this.) A general purpose slow release fertilizer applied each spring should be adequate (look for 10-10-10 plus minors), this is not a fussy plant. Keep the soil evenly moist during the growing season, like a wrung out sponge.
As far as winter care, you might leave it outdoors as long as possible in the fall, protecting it on cold nights and setting it outside during the day. This will help it to adjust to the season change and slow down for the winter yet extend its outdoor growing season a bit. Then when temperatures begin to fall below 50, allow it to rest in a cool sunny room at about 45 degrees. Water lightly so it does not dry out and do not fertilize while it is resting. Then gradually reacclimate it to being outside in the spring so it reawakens naturally with the season.
The agapanthus should respond well to the same type of winter care, but give it full sun and a more generous fertilization program during the summer, also make sure it receives ample water when it is growing vigorously.
I hope this helps!
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