https://file.aiquickdraw.com/mathbot/file/1725764014277wr1atm70.jpg

The image contains a graph of a function \( y = f(x) \) and a series of questions about the function's properties.

Let's analyze each part step by step:

1. **Question (a):**  
   Is \( f(-2) \) negative?  
   To answer this, we need to locate \( x = -2 \) on the graph and see if the function value is below the x-axis. From the graph, \( f(-2) \) appears to be negative because the graph is below the x-axis at \( x = -2 \).

   **Answer:** Yes, \( f(-2) \) is negative.

2. **Question (b):**  
   For which value(s) of \( x \), is \( f(x) = 0 \)?  
   To find where \( f(x) = 0 \), we need to check the points where the graph intersects the x-axis. From the graph, the function appears to cross the x-axis at three points: \( x = -3 \), \( x = 0 \), and \( x = 2 \).

   **Answer:** \( x = -3, 0, 2 \).

3. **Question (c):**  
   For which value(s) of \( x \), is \( f(x) \leq 0 \)?  
   This asks for the intervals where the function is either on or below the x-axis. From the graph, the function is below or equal to zero from \( x = -3 \) to \( x = 0 \) and from \( x = 2 \) onwards. Using interval notation:

   **Answer:** \( x \in [-3, 0] \cup [2, \infty) \).

Would you like further details or explanations on any part?  
Here are five related questions you might find helpful:

1. What is the significance of \( f(x) = 0 \) in real-world applications?
2. How can you determine if a function is increasing or decreasing from a graph?
3. What does it mean for a function to have a negative value at a particular \( x \)-value?
4. How do you express intervals where a function is negative using interval notation?
5. How does the graph of a function help in identifying its domain and range?

**Tip:**  
When analyzing a function graph, always check for x-axis intersections (roots), where the function is above or below the x-axis, and how it behaves at its critical points (maximums and minimums).

Use the graph of the function y = f(x) to answer the questions:
(a) Is f(-2) negative?
(b) For which value(s) of x is f(x) = 0?
(c) For which value(s) of x is f(x) ≤ 0? Write your answer using interval notation.

Explore how to analyze the graph of a function y = f(x) to determine the sign of the function at specific points, find where the function equals zero, and identify intervals where the function is non-positive. Suitable for students in Grades 9-12.

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