A population is modeled by 
f(x) = 6 · 3x.  Is the average rate of change for f constant or does it vary

The function given is \( f(x) = 6 \cdot 3^x \), which is an **exponential function**.

To determine whether the **average rate of change** of this function is constant or variable, let's analyze its nature.

### Average Rate of Change:
The **average rate of change** of a function \( f(x) \) over an interval \( [a, b] \) is given by:

\[
\text{Average Rate of Change} = \frac{f(b) - f(a)}{b - a}
\]

For exponential functions like \( f(x) = 6 \cdot 3^x \), the rate of change depends on the values of \( a \) and \( b \) chosen. This is because exponential functions grow by a constant factor (in this case, by a factor of 3) rather than by a constant amount.

### Conclusion:
The **average rate of change** of \( f(x) = 6 \cdot 3^x \) **varies** and is not constant. This is a characteristic of exponential functions, which do not have a constant rate of change like linear functions.

Would you like more details on how the rate of change varies for this function?

Here are 5 related questions:
1. How do you calculate the average rate of change for any function?
2. What is the difference between linear and exponential functions in terms of growth?
3. How do you determine the instantaneous rate of change for exponential functions?
4. Can exponential functions model real-world population growth effectively?
5. What is the impact of the base (3 in this case) on the growth rate of the function?

**Tip:** For exponential growth, the base of the exponent (like 3 here) indicates the multiplication factor for each unit increase in the input.

A population is modeled by f(x) = 6 · 3^x. Is the average rate of change for f constant or does it vary?

Explore the behavior of exponential functions with f(x) = 6 · 3^x and determine if the average rate of change is constant or variable. This problem introduces key concepts such as exponential growth, the average rate of change formula, and how growth varies over intervals. Suitable for Grades 9-12 students studying advanced algebra.

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