1. **State the problem:** We need to find the slope of the tangent line to the curve at the point $(-1,0)$. The slope of the tangent line represents the derivative of the function at that point. 2. **Identify the function:** Since the graph is not explicitly given, but the tangent line touches the curve at $(-1,0)$, we assume the function is differentiable there. 3. **Use the definition of slope of tangent:** The slope $m$ of the tangent line at $x=-1$ is given by the derivative $f'(x)$ evaluated at $x=-1$. 4. **Analyze the options:** The possible slopes are $-1$, $-\frac{1}{2}$, $-2$, and $-4$. 5. **Interpret the graph:** The tangent line at $(-1,0)$ is shown in blue. By visual inspection, the tangent line appears to be steep and descending. 6. **Estimate slope:** The slope is the "rise over run". From the graph, the tangent line seems to drop 2 units vertically for every 1 unit it moves horizontally to the right, indicating a slope of $-2$. 7. **Final answer:** The slope of the tangent line at $(-1,0)$ is $\boxed{-2}$.