Completing the square gives
[tex]\dfrac14x^2+bx+10=\dfrac14(x^2+4bx+40)=\dfrac14(x^2+4bx+4b^2+40-4b^2)[/tex]
[tex]\implies\dfrac14x^2+bx+10=\dfrac14(x+2b)^2+10-b^2[/tex]
At the point on the graph along the axis of symmetry, the squared term vanishes, so that when [tex]x=6[/tex], we have [tex]x+2b=0[/tex]. So
[tex]6+2b=0\implies 2b=-6\implies\boxed{b=-3}[/tex]
