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Jack works for a company that pays him $20 an hour. His normal workday does not exceed 8 hours. After working 8 hours in a day, he gets paid $5 per hour overtime in addition to his regular wage. He cannot work fewer than 5 hours a day or more than 14 hours a day. Which graph shows Jack's wages if he works for *x* hours a day?

The question describes Jack’s wage based on regular and overtime pay, and the correct graph will reflect the change in his hourly rate when overtime starts (after 8 hours). Let's analyze the wage function step by step:

1. **Regular Pay**:  
   - For the first 8 hours, Jack earns **$20 per hour**.  
   - So, for $x \leq 8$, his total wage is $20x$.

2. **Overtime Pay**:  
   - For hours beyond 8, he earns $20 per hour **plus** an extra $5, making it **$25 per hour**.  
   - So, for $x > 8$, the wage function becomes:
     \[ 20(8) + 25(x - 8) = 160 + 25(x - 8) \]

3. **Work Constraints**:
   - Jack works between **5 to 14 hours a day**. The graph should start from $x = 5$ and end at $x = 14$.

### Solution Approach
- The graph should have a **piecewise linear structure**:
  - A straight line with a slope of 20 from $x = 5$ to $x = 8$.
  - Then, a steeper slope (25) starting from $x = 8$ to $x = 14$.

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### Identifying the Correct Graph
After inspecting the graphs:
- **Graph A** shows a **clear increase in slope** after 8 hours, indicating overtime pay. This matches the piecewise nature of Jack’s wage function.  
  - For $x \leq 8$, the line rises more gently (regular wage).
  - For $x > 8$, the line becomes steeper (overtime wage).

### Conclusion
The correct graph representing Jack's daily wages is **Graph A**.

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Do you need a more detailed explanation or further clarifications? 

Here are five follow-up questions you might explore:
1. How would the graph change if the overtime rate increased to $30 per hour?
2. How can we represent the wage function algebraically as a piecewise function?
3. What happens if Jack only works between 6 and 10 hours daily?
4. How does the slope of the line relate to Jack’s hourly rate in general?
5. What if Jack gets a fixed bonus added to his daily wage—how would that alter the graph?

**Tip:** When dealing with wage functions, always check for any shifts in the slope to identify changes in pay rate.

Jack works for a company that pays him $20 an hour. His normal workday does not exceed 8 hours. After working 8 hours in a day, he gets paid $5 per hour overtime in addition to his regular wage. He cannot work fewer than 5 hours a day or more than 14 hours a day. Which graph shows Jack's wages if he works for x hours a day?

Explore how to determine the correct graph representing Jack's daily wage based on regular and overtime pay. Understand piecewise functions, linear equations, and the relationship between hours worked and wages. Suitable for students in Grades 9-12.

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