Let's go through each question based on the graph shown.

Solution

1. Question (a): Find $f(1)$.

   • From the graph, locate $x = 1$ on the $x$-axis and check the value of $f(x)$ at that point. It appears that $f(1) \approx 2$.

2. Question (b): Solve $f(x) = 0$ (find all $x$-values where $f(x) = 0$).

   • $f(x) = 0$ where the graph crosses the $x$-axis. Based on the graph, the function intersects the $x$-axis at two points:
     • $x \approx -2.5$
     • $x = 0$
   • Thus, the solutions to $f(x) = 0$ are approximately $x = -2.5$ and $x = 0$.

3. Question (c): Find the local extrema of $f$.

   • Local extrema are points where the function reaches a local maximum or minimum. Observing the graph:
     • There is a local maximum at approximately $x = -1$ with $f(x) \approx 3$.
     • There is a local minimum at approximately $x = 0.5$ with $f(x) \approx -1$.

Summary of Answers

• (a) $f(1) \approx 2$.
• (b) The solutions to $f(x) = 0$ are $x \approx -2.5$ and $x = 0$.
• (c) Local maximum at $x \approx -1$ and local minimum at $x \approx 0.5$.

Would you like further details on any specific part?

Related Questions

1. How can we confirm the exact values for the local extrema using calculus?
2. What is the significance of finding $f(x) = 0$ in understanding the function's behavior?
3. How would you describe the end behavior of this function?
4. What could be the possible degrees of the polynomial function based on the graph?
5. How can we determine the intervals where the function is increasing or decreasing?

Tip

When analyzing a graph, pay close attention to the points where the function intersects the axes and any turning points; these are often critical points for solving various types of function-based problems.