To identify exponential growth and decay relationships, we need to recognize the general forms:

1. Exponential Growth: $y = a \cdot b^x$, where $a > 0$ and $b > 1$.
2. Exponential Decay: $y = a \cdot b^x$, where $0 < b < 1$.

Let's analyze each equation:

• A: $y = x^4$ is a polynomial function, not exponential.
• B: $y = x^{-4}$ is also a polynomial function, not exponential.
• C: $y = 3 \cdot 4^x$ follows the form $y = a \cdot b^x$ with $a = 3$ and $b = 4$, which is an exponential growth because $b = 4 > 1$.
• D: $y = 3 \cdot (4^{-x})$ can be rewritten as $y = 3 \cdot \left(\frac{1}{4}\right)^x$, which fits the form of exponential decay because $0 < \frac{1}{4} < 1$.

Answer:

• Exponential decay: D: $y = 3 \cdot 4^{-x}$
• Exponential growth: C: $y = 3 \cdot 4^x$

Would you like further details or have any questions?

Here are five related questions to expand on this:

1. How can you differentiate between linear and exponential relationships in general?
2. What are common real-life examples of exponential growth and decay?
3. How does changing the base $b$ affect the rate of exponential growth or decay?
4. Can a function switch between exponential growth and decay, and if so, how?
5. What is the significance of the initial value $a$ in an exponential function?

Tip: In exponential functions, if the base is between 0 and 1, it represents decay; if greater than 1, it represents growth.