point: a dot on your graph indicating that this combination of leisure/income is on the constraint
hollow point: a little circle on your graph indicating that all points up this this one are on the budget constraint, but the point itself is not (your professor may use a different notation for this)
The way i think about these is to start with 100% of leisure, then think about what happens if you spend some more time working. Keep repeating this until you have 100% labour.
Ill assume there are 30 days in the month, so the maximum leisure is 30*24=720. This corresponds to an income of 0.
for the first 120 hours, any additional work will credit him with $10 an hour. So you have another point at:
leisure: 720 - 120 =600
This is a hollow point because if you get there you will immediately be given the $600 allowance.
Draw a stright line between them (because income/leisure are swapped at constant rate of $10 per hour)
Now, at this point, he becomes eligible for the $600 grant; so we have a discontinuity. Draw another point:
income $1200 + 600=$1800
Dont make the mistake of drawing a vertical line between the 2 points you just drew, this is not correct (presumably, you must take either the entire grant or none of it, so the points in between are not achievable).
Now. assume the upper limit on earnings is $2500 (as in the quesion). i assume this means $2500 labour income, not $2500 total income (which would include the grant).
To get the $2500 labour income, you'd have to work 250 hours, which is an additional 130 hours). So, up to that point our parent is just swapping leisure for income at the rate of $10/hour again:
leisure: 600-130 = 470
Draw a line connecting the previous two points, because every space in between is acheivable.
But, he is no longer eligible for the grant, so his income drops by 600:
Finally, there is no more assistance available after this, so the final point you must draw is 100% labour.
income: 2500 + 470*10 =7200
Ill see if i can draw it and attach in another reply.