Show that the heat equation $u_t = u_{xx}$ has solutions of the form $u(x, t) = f (x − at)$ where $a$ is a constant. Show that $a$ can have any value, real or complex, and describe the behavior of the solutions in both cases.
I'm a little unsure of what this question is looking for? Because when I plug f into the heat equation I'm left with $-af_t=f_{xx}$
Any guidance would be appreciated
HINT
To elaborate on the discussion in the messages, let $u(x,t) = f(x-at)$ for some twice differentiable function $f$ and a constant $a \in \mathbb{C}$. Note that $$ u_t = \frac{\partial u}{\partial t} = \frac{\partial f(x-at)}{\partial t} = f'(x-at) \frac{\partial [x-at]}{\partial t} = -a f'(x-at). $$ Can you use a similar technique (Chain Rule, really) to compute $u_x, u_{xx}$ and verify that this form of $u$ indeed satisfies the desired equation?