https://file.aiquickdraw.com/mathbot/file/1725478830706ekbrfa0q.jpg

The graph shows the function \( y = f(x) \) with important points and features marked. Let's break down and solve each part of the problem based on the graph.

### 1. Find \( f(2) \)
Looking at the graph, when \( x = 2 \), the corresponding \( y \)-value (point on the graph) is \( -1 \). Therefore:
\[
f(2) = -1
\]

### 2. Find all the \( x \)-values such that \( f(x) = 0 \)
To find where \( f(x) = 0 \), we look for points where the graph crosses the x-axis. The graph intersects the x-axis at the following points:
\[
x = -3, x = -1, x = 0, x = 3
\]
So, the \( x \)-values where \( f(x) = 0 \) are \( -3, -1, 0, 3 \).

### 3. Find the range of \( f \)
The range of a function is the set of possible \( y \)-values. From the graph, the function reaches a maximum at \( y = 3 \) (at \( x = 4 \)) and a minimum at \( y = -3 \) (at \( x = -4 \)). Therefore, the range of \( f \) is:
\[
\text{Range of } f = [-3, 3]
\]

### 4. Find all local extrema of \( f \)
Local extrema are the local minimum and maximum points. By observing the graph:
- There is a local minimum at \( (-4, -3) \)
- There is a local maximum at \( (-1, 1) \)
- There is a local minimum at \( (2, -1) \)

So, the local extrema are:
- Local minimum at \( (-4, -3) \)
- Local maximum at \( (-1, 1) \)
- Local minimum at \( (2, -1) \)

### 5. Find all absolute extrema of \( f \)
Absolute extrema are the highest and lowest points on the entire graph. From the graph:
- The absolute minimum is at \( (-4, -3) \)
- The absolute maximum is at \( (4, 3) \)

So, the absolute extrema are:
- Absolute minimum: \( (-4, -3) \)
- Absolute maximum: \( (4, 3) \)

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Do you want more detailed explanations or have any specific questions on these results?

Here are some related questions to consider:
1. What happens to the function as \( x \to -\infty \) based on the arrow at \( x = -4 \)?
2. How do you determine if a point is a local maximum or minimum?
3. What is the difference between absolute and local extrema?
4. How does the closed endpoint at \( (4, 3) \) influence the range of the function?
5. Can the function have additional extrema beyond what is shown in the graph?

**Tip**: When finding the range of a function, always check the endpoints and any maxima or minima within the interval.

Consider the graph of the function y = f(x) given below. Based on the graph, answer the following questions:
1. Find f(2).
2. Find all the x-values such that f(x) = 0.
3. Find the range of f.
4. Find all local extrema of f.
5. Find all absolute extrema of f.

This problem involves analyzing the graph of a function to find specific values such as f(2), the x-values where f(x) = 0, the range of the function, and identifying both local and absolute extrema. The problem is suitable for students in Grades 10-12, focusing on key concepts such as graphing, functions, and extrema.

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