A Bounded Quadratic Bounds a Line

Let $a,b,c\in\mathbb R$, and consider $f(x)=ax^2+bx+c$ and $g(x)=ax+b$, both on $-1\leqslant x\leqslant 1$. Prove that if $|f(x)|\leqslant 1$ for all $x\in[-1,1]$, then $|g(x)|\leqslant 2$ for all $x\in[-1,1]$.