You can use Lagrange polynomial interpolation, the curve is given by
y(x) = y_1 * (x-x_2)*(x-x_3)/((x_1-x_2)*(x_1-x_3))
+ y_2 * (x-x_1)*(x-x_3)/((x_2-x_1)*(x_2-x_3))
+ y_3 * (x-x_1)*(x-x_2)/((x_3-x_1)*(x_3-x_2))
If you collect the coefficients, you obtain
a = y_1/((x_1-x_2)*(x_1-x_3)) + y_2/((x_2-x_1)*(x_2-x_3)) + y_3/((x_3-x_1)*(x_3-x_2))
b = -y_1*(x_2+x_3)/((x_1-x_2)*(x_1-x_3))
-y_2*(x_1+x_3)/((x_2-x_1)*(x_2-x_3))
-y_3*(x_1+x_2)/((x_3-x_1)*(x_3-x_2))
c = y_1*x_2*x_3/((x_1-x_2)*(x_1-x_3))
+ y_2*x_1*x_3/((x_2-x_1)*(x_2-x_3))
+ y_3*x_1*x_2/((x_3-x_1)*(x_3-x_2))