to make the function continuous, you need to make sure that at each boundary of the ranges, so when the piece wise function changes, that the y is still the same, that means that for example, at x = 3, ax2 - bx + 3 = 2x - a + b. This is because if you graph these two "mini functions" they must meet at the same point. if the y-value for x=3 of both functions is different, the graph will have a gap.
you need to solve for a and b so you have two variables and need two equations.
you need to ensure continuity at two points where the middle range stops and starts: 2, 3
to set continuity at x = 2
x - 2 = ax2 - bx + 3
which is re written as
2 - 2 = a(2)2 - b*2 + 3
to set continuity at x = 3
ax2 - bx + 3 = 2x - a + b
which is re written as
a(3)2 - b*3 + 3 = 2*3 - a + b
solving for a and b with simple algebra leads to
a = 9/2
b = 21/2
(note: due to the formating it's unclear which function is for x < 2. I read it as "x - 2" but if that isn't what the question says, replace "x - 2" with the actual function for x < 2)
