https://file.aiquickdraw.com/mathbot/file/1726790247494weqvuia4.jpg

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 \( x = 15 \)).

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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.

Use the graph of the function f to find approximations of the given values. The questions are: (a) f(-5), (b) f(5), (c) f(15), and (d) (f(15) - f(5))/(15 - 5).

Learn how to approximate function values from a graph and calculate the difference quotient for a given interval. This problem involves analyzing a function's graph, understanding key concepts like average rate of change, and applying the slope formula. Suitable for students in Grades 10-12.

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