Gradient Estimate D14Edc

1. **State the problem:** We need to estimate the gradient of the curve $y = f(x)$ at the point where $x = -6$ using the tangent line drawn at that point. 2. **Recall the concept:** The gradient of the curve at a point is the slope of the tangent line to the curve at that point. 3. **Estimate the slope of the tangent line:** To find the slope, we use the formula for the gradient of a line: $$m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1}$$ 4. **Identify two points on the tangent line:** From the graph, approximate two points on the tangent line near $x = -6$. Suppose the points are approximately $(-7, 5)$ and $(-5, 1)$. 5. **Calculate the gradient:** $$m = \frac{1 - 5}{-5 - (-7)} = \frac{-4}{2} = -2$$ 6. **Interpretation:** The estimated gradient of the curve at $x = -6$ is $-2$. **Final answer:** The gradient of the curve at $x = -6$ is approximately $-2$.