https://file.aiquickdraw.com/mathbot/file/17296287156791dvy5g6o.jpg

https://file.aiquickdraw.com/mathbot/file/1729628715679f3ncuofn.jpg

The two graphs you uploaded represent the functions:

- \( y = 5x \) (linear growth) on the left
- \( y = 5^x \) (exponential growth) on the right

Let's break down the growth between \( x = 0 \) and \( x = 2 \):

1. **For \( y = 5x \)**: This is a linear function where the value of \( y \) increases by 5 for each unit increase in \( x \). 
   - At \( x = 0 \), \( y = 0 \)
   - At \( x = 2 \), \( y = 10 \)
   The growth is constant and predictable.

2. **For \( y = 5^x \)**: This is an exponential function, where the value of \( y \) increases by multiplying 5 for each unit increase in \( x \). 
   - At \( x = 0 \), \( y = 5^0 = 1 \)
   - At \( x = 2 \), \( y = 5^2 = 25 \)
   The growth rate increases more rapidly.

### Comparing Growth:
From \( x = 0 \) to \( x = 2 \), the exponential function \( y = 5^x \) grows faster than the linear function \( y = 5x \). This is evident because \( y = 5x \) reaches 10 at \( x = 2 \), whereas \( y = 5^x \) reaches 25.

### Correct Answer:
- \( y = 5x \) grows **slower** than \( y = 5^x \).

### Would you like further details or have any questions?

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Here are five related questions to extend your understanding:

1. How does the slope of a linear function like \( y = 5x \) relate to its growth?
2. What would happen to the exponential growth if the base of the exponent changed from 5 to 2 in \( y = 5^x \)?
3. Can you determine the point where \( y = 5^x \) overtakes \( y = 5x \) for larger values of \( x \)?
4. How does the concept of "exponential growth" apply in real-world scenarios like population growth?
5. How would the graph change if the linear function had a different slope, such as \( y = 2x \)?

**Tip:** When comparing functions, always pay attention to whether they exhibit linear, exponential, or logarithmic growth patterns, as these grow at very different rates over time.

From x = 0 to x = 2, which of the following best describes the growth of the two functions below?

Compare the growth of the linear function y = 5x and the exponential function y = 5^x between x = 0 and x = 2. Discover how exponential growth outpaces linear growth, with y = 5x reaching 10 and y = 5^x reaching 25 at x = 2.

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