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For the graph of the function y = f(x), find the absolute maximum and the absolute minimum, if it exists.


From the graph of the function \( y = f(x) \), we can identify several critical points and observe where the function reaches its highest and lowest values.

### Critical points from the graph:
- At \( x = -1 \), \( f(-1) = 3 \).
- At \( x = 1 \), \( f(1) = 2 \).
- At \( x = 4 \), \( f(4) = 7 \) (highest point).
- At \( x = 6 \), \( f(6) = 5 \).

### Absolute Maximum:
- The function reaches its highest value at \( (4, 7) \). Therefore, the **absolute maximum** is \( 7 \) at \( x = 4 \).

### Absolute Minimum:
- The lowest value of the function occurs at \( (1, 2) \). Therefore, the **absolute minimum** is \( 2 \) at \( x = 1 \).

### Summary:
- **Absolute maximum**: \( 7 \) at \( x = 4 \).
- **Absolute minimum**: \( 2 \) at \( x = 1 \).

Would you like more details or have any questions? 

Here are some related questions to deepen your understanding:
1. How would you identify local maximum and minimum points from the graph?
2. What is the difference between local and absolute extrema in a function?
3. How can you verify the critical points of a function algebraically?
4. What role do endpoints play when determining the absolute extrema on a closed interval?
5. How would the graph change if it had vertical asymptotes?

**Tip**: In a graph, the absolute maximum and minimum are typically found by analyzing the peaks and troughs, but verifying algebraically is always recommended when possible.

For the graph of the function y = f(x), find the absolute maximum and the absolute minimum, if it exists.

Learn how to find the absolute maximum and minimum points on a graph of the function y = f(x) by analyzing critical points and extrema. This guide provides a clear explanation of identifying these points with examples.

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