reflection of linear function

1. Understand the transformation: The problem asks for the graph of $-f(x)$ given the function $f(x) = x + 1$. In function transformations, the negative sign outside the function, $-f(x)$, represents a reflection of the original graph across the $x$-axis.

1. Determine the new equation: Since $f(x) = x + 1$, then $-f(x) = -(x + 1) = -x - 1$. This new function is a line with a slope of $-1$ and a $y$-intercept of $-1$.

3. Analyze the options:
- The original function $f(x) = x + 1$ passes through $(-1, 0)$ and $(0, 1)$.
- Reflecting these points across the $x$-axis gives $(-1, 0)$ and $(0, -1)$.
- We look for the line that passes through $(0, -1)$ and $(-1, 0)$.

4. Identify the correct graph:
- Option 1 passes through $(1, 0)$ and $(0, -1)$. This is $y = x - 1$.
- Option 2 passes through $(-1, 0)$ and $(0, -1)$. This matches our derived equation $y = -x - 1$.
- Option 3 passes through $(1, 0)$ and $(0, 1)$. This is $y = -x + 1$.
- Option 4 is the original function $f(x) = x + 1$.

Therefore, Option 2 is the correct graph.