https://file.aiquickdraw.com/mathbot/file/1727985241774ly6usgqj.jpg

The graphs provided compare the functions \( y = 4x \) (linear) and \( y = 4^x \) (exponential).

### Analysis of Growth:
- **Linear Function: \( y = 4x \)** grows at a constant rate. This is shown by the straight line in blue, indicating a steady, predictable increase as \( x \) increases.
- **Exponential Function: \( y = 4^x \)** grows at an increasing rate, starting slower but eventually surpassing the linear function as \( x \) gets larger. The green curve in the graph shows this rapid growth.

### General Growth Comparison:
- At first, \( y = 4x \) grows faster (as seen for lower values of \( x \)). But as \( x \) increases, the exponential function \( y = 4^x \) eventually grows faster than \( y = 4x \).
  
### Conclusion:
The correct statement is: **The function \( y = 4^x \) is growing faster than \( y = 4x \)**.

Would you like further details on why exponential functions grow faster over time compared to linear functions? 

Here are five related questions to consider:
1. How do we mathematically compare growth rates of linear vs exponential functions?
2. At what exact point does the function \( y = 4^x \) surpass \( y = 4x \)?
3. What are the key differences between linear and exponential growth?
4. How does the base of the exponential function (4 in \( 4^x \)) affect the speed of growth?
5. How would the graphs change if we used different constants in the linear or exponential equations?

**Tip**: Exponential growth eventually overtakes linear growth, no matter how large the slope of the linear function is. This is a key feature of exponential functions.

In general, how does the growth of y = 4^x compare to the growth of y = 4x?

Explore how the exponential function y = 4^x grows compared to the linear function y = 4x. This comparison helps understand how exponential growth eventually overtakes linear growth. Suitable for students in Grades 9-12.

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