f(x) = 6 · 3x what is the percentage rate of change of f between
x = 3
and
x = 4

To calculate the percentage rate of change of the function \( f(x) = 6 \cdot 3^x \) between \( x = 3 \) and \( x = 4 \), we can follow these steps:

1. **Evaluate \( f(3) \):**
   \[
   f(3) = 6 \cdot 3^3 = 6 \cdot 27 = 162
   \]

2. **Evaluate \( f(4) \):**
   \[
   f(4) = 6 \cdot 3^4 = 6 \cdot 81 = 486
   \]

3. **Find the absolute change in \( f(x) \):**
   \[
   \Delta f = f(4) - f(3) = 486 - 162 = 324
   \]

4. **Find the percentage rate of change:**
   The formula for percentage rate of change is:
   \[
   \text{Percentage change} = \frac{\Delta f}{f(3)} \times 100 = \frac{324}{162} \times 100 = 200\%
   \]

Thus, the percentage rate of change of \( f(x) \) between \( x = 3 \) and \( x = 4 \) is **200%**.

Do you need more details or have any questions?

Here are 5 related questions for deeper understanding:
1. How would the percentage rate of change differ for smaller intervals, say between \( x = 2 \) and \( x = 3 \)?
2. What is the derivative of \( f(x) = 6 \cdot 3^x \), and how does it relate to the rate of change at any \( x \)?
3. How does the base of the exponential function, 3, influence the growth rate of the function?
4. What would happen if the base were a fraction, say \( f(x) = 6 \cdot \left(\frac{1}{3}\right)^x \)?
5. How can we interpret the 200% growth in real-world applications?

**Tip:** In exponential functions, the base dictates how quickly the function grows or decays.

f(x) = 6 · 3^x what is the percentage rate of change of f between x = 3 and x = 4?

Explore how to find the percentage rate of change of the exponential function f(x) = 6 · 3^x between x = 3 and x = 4. This step-by-step solution covers the evaluation of f(x), the absolute change, and the final percentage growth calculation. Suitable for students in Grades 9-11.

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