The image shows a graph of a function $f$ and a set of questions asking for approximations of the function at specific values of $x$, as well as the difference quotient (or average rate of change) between two points.

Let's break it down step by step based on the graph and the prompts:

1. (a) $f(-5)$:

   • From the graph, locate $x = -5$ on the horizontal axis.
   • Approximate the corresponding value of $y$, which gives $f(-5)$.
   • It looks like $f(-5) \approx 5$.

2. (b) $f(5)$:

   • Locate $x = 5$ on the graph.
   • Approximate the corresponding $y$-value, which gives $f(5)$.
   • It seems that $f(5) \approx 15$.

3. (c) $f(15)$:

   • Locate $x = 15$ on the graph.
   • Approximate the corresponding $y$-value, which gives $f(15)$.
   • The function at $x = 15$ is approximately $f(15) \approx 15$.

4. (d) $\frac{f(15) - f(5)}{15 - 5}$:

   • Using the values approximated above: $f(15) \approx 15$ and $f(5) \approx 15$.
   • Calculate the difference quotient: $\frac{f(15) - f(5)}{15 - 5} = \frac{15 - 15}{10} = 0$
   • The result is 0, indicating that the slope of the line between these two points is 0 (the function is flat between $x = 5$ and $x = 15$).

Would you like more detailed explanations or have any further questions? Here are some related questions for further practice:

1. What is the slope of the line between $f(-5)$ and $f(5)$?
2. Can you determine where the function is increasing or decreasing based on the graph?
3. How would you estimate $f(0)$ from the graph?
4. What is the average rate of change of the function between $f(10)$ and $f(20)$?
5. How can you tell if a function is concave up or concave down from its graph?

Tip: When interpreting graphs, pay close attention to the trends between points to understand the function's behavior—whether it's increasing, decreasing, or staying constant.