A Min Range of a Shifting Quadratic

Let $f(x)=ax^2-2016x+2017$ ($a>0$). On the interval $[t-1,t+1]$, let $M(t)$ and $m(t)$ be the maximum and minimum of $f(x)$, and let $h(t)=M(t)-m(t)$. If the minimum value of $h(t)$ is $1$, find $a$.