The problem asks to find the slope of the tangent line to the function $f(x)$ at $x = 3$, based on the graph provided.

Step-by-step solution:

1. Identify the point of tangency: The point where $x = 3$ is marked on the curve. From the graph, you can visually estimate the corresponding $y$-coordinate.

2. Draw the tangent line: A tangent line is a straight line that touches the curve at exactly one point. By placing a point at $x = 3$ on the graph and visualizing the tangent, we can observe the line’s steepness or direction.

3. Calculate the slope of the tangent line: The slope of a tangent line is equivalent to the derivative of the function at that point, $f'(3)$. From the graph, you can estimate two points on the tangent line (e.g., $(3, y_1)$ and another point $(x_2, y_2)$ on the line).

   The slope $m$ is calculated using: $m = \frac{y_2 - y_1}{x_2 - x_1}$ where $(x_1, y_1)$ and $(x_2, y_2)$ are points on the tangent line.

   From the graph, it appears that the slope is negative (the tangent line slopes downward as $x$ increases). Using visual estimation, you can approximate the values and calculate the slope.

Would you like more details or further clarification on any steps?

Expanding concepts:

1. How do we calculate the slope of a line if given two points?
2. What is the significance of a tangent line in calculus?
3. How do you find the derivative of a function at a specific point?
4. Can the slope of the tangent line be different from the slope of the secant line?
5. What are the applications of tangent lines in real-world problems?

Tip:

Remember that the slope of the tangent line can be approximated graphically, but for an exact value, you would typically need the function $f(x)$ and its derivative.