https://file.aiquickdraw.com/mathbot/file/1726832105905v21qwibj.jpg

The problem involves analyzing the graph of the function \( I = f(w) \), where \( I \) represents the number of individuals (in thousands) infected with influenza, and \( w \) represents the number of weeks after the epidemic began.

### Analyzing the Graph:

The graph provided shows the function \( f(w) \) over time:

- **X-axis (horizontal)**: Represents weeks \( w \) after the epidemic began.
- **Y-axis (vertical)**: Represents the number of infected individuals \( I \) in thousands.

The question asks to estimate the values of \( w \) where \( f(w) = 4.5 \).

### Finding \( w \) for \( f(w) = 4.5 \):

1. **Locate 4.5 on the Y-axis**: Identify the point on the Y-axis that corresponds to 4.5 (thousand) infected individuals.
  
2. **Draw a horizontal line at \( I = 4.5 \)**: From \( I = 4.5 \), draw a horizontal line across the graph until it intersects the curve \( f(w) \).

3. **Determine the intersection points**: Find the corresponding values of \( w \) at these intersection points. 

### Estimate from the Graph:
- The graph shows that \( f(w) = 4.5 \) occurs approximately around **week 3** and **week 13**.

### Interpretation in Terms of the Epidemic:

- **At week 3**: The number of infected individuals reached 4.5 thousand as the epidemic was increasing.
- **At week 13**: The number of infected individuals decreased back to 4.5 thousand as the epidemic was subsiding.

This indicates that the epidemic peaked somewhere between weeks 3 and 13 and then started to decline.

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Would you like more details on how to interpret the graph, or do you have any other questions?

### Further Questions:
1. How can you estimate other values of \( w \) for different infection levels from this graph?
2. What does the peak of the curve represent in the context of the epidemic?
3. How could you use this graph to estimate when the epidemic started to decline?
4. What does the steepness of the curve tell you about the rate of infection?
5. If you had the equation of the curve, how would you solve for exact values of \( w \)?

**Tip:** When interpreting graphs, always consider the units and scale for each axis to ensure accurate analysis.

For approximately which w's is f(w) = 4.5? Explain what the estimates mean in terms of the epidemic.

Learn how to interpret a graph representing an epidemic to estimate the weeks where f(w) = 4.5, representing 4.5 thousand individuals infected. This problem teaches graph reading skills and understanding function behavior over time. Suitable for students in Grades 8-10.

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