The standard equation for a parabola is y=Ax^2+Bx+C.
If we substitute the x values for the given points with their y values we get:
(1) 6=A+B+C; (2) 17=4A+2B+C; (3) 22=9A-3B+C
(2)-(1): (4) 11=3A+B
(2)-(3): (5) -5=-5A+5B; 1=A-B
(4)+(5): 12=4A, so A=3; therefore B=11-9=2; and C=6-(A+B)=1
y=3x^2+2x+1