Consider the following relationships:

A: y=x4

B: y=x−4

C: y=34x

D: y=3−4x

1. Which of these is an example of an exponential decay relationship and which is an example of an exponential growth relationship?

Exponential decay: 
(No answer given)

Exponential growth:

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.

Consider the following relationships:

A: y = x^4

B: y = x^{-4}

C: y = 3 * 4^x

D: y = 3 * 4^{-x}

1. Which of these is an example of an exponential decay relationship and which is an example of an exponential growth relationship?

Understand how to identify exponential growth and decay relationships by analyzing functions such as y = 3 * 4^x and y = 3 * 4^{-x}. Learn the key formulas for exponential functions, explore the difference between polynomial and exponential forms, and recognize the signs of growth and decay. Suitable for students in Grades 9-12.

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