Here are the graphs of the forward-moving wave, backward-moving wave, and the solution at the given times:

1. For $t = 0.25$:

   • The forward-moving wave is shifted right by 0.25.
   • The backward-moving wave is shifted left by 0.25.
   • The solution shows a non-zero overlap from $-0.75 < x < 0.75$.

2. For $t = 0.75$:

   • The forward-moving wave is shifted right by 0.75.
   • The backward-moving wave is shifted left by 0.75.
   • The solution shows a non-zero overlap from $-0.25 < x < 0.25$.

3. For $t = 2$:

   • The forward-moving wave is shifted right by 2.
   • The backward-moving wave is shifted left by 2.
   • There is no overlap between the waves, resulting in a zero solution everywhere.

Would you like more details or further clarification on any part of this process?

Related Questions:

1. How would the solution change if the initial velocity $u_t(x, 0)$ were not zero?
2. Can you explain the effect of different wave speeds on the solution of the wave equation?
3. How does D'Alembert's solution extend to multi-dimensional problems?
4. What is the physical significance of forward and backward-moving waves?
5. How would the solution change if $f(x)$ were a different shape, such as a Gaussian?

Tip: In wave equations, the overlap of forward and backward-moving waves determines the shape of the solution at any given time.