To determine $f'(a)$ for the given function $f(x)$ at specific points, we need to estimate the slope of the tangent line to the curve at those points. The slope of the tangent line represents the derivative, $f'(x)$, at that point.

Let's analyze the graph provided for each value of $a = 1, 2, 4, 7$:

Step-by-step Analysis:

1. At $a = 1$:

   • The graph is relatively flat at $x = 1$.
   • The slope of the tangent line at this point appears to be close to zero.
   • Therefore, $f'(1) \approx 0$.

2. At $a = 2$:

   • The graph is increasing linearly at $x = 2$.
   • The slope of the tangent line appears to be constant and positive.
   • If we estimate the slope by using two points on this line, we get a positive slope.
   • So, $f'(2) \approx 1$.

3. At $a = 4$:

   • At $x = 4$, the graph reaches its maximum point.
   • The slope of the tangent line at the maximum point is horizontal.
   • Therefore, $f'(4) = 0$.

4. At $a = 7$:

   • The graph is steeply decreasing at $x = 7$.
   • The slope of the tangent line appears to be negative and quite steep.
   • An approximate value for the negative slope could be around $-2$.
   • So, $f'(7) \approx -2$.

Conclusion: