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 $x = 14$.

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.

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.