1977 AMC 12 Question 22

If $f(x)$ is a real-valued function of the real variable $x,$ and $f(x)$ is not identically zero, and for all $a$ and $b$ $$
\begin{gathered} f(a+b)+f(a-b)\\ =2f(a)+2f(b), \end{gathered}
$$ then for all $x$ and $y$