A variety of functions are shown below.

Complete the table to determine the percent rate of change of each function.  Choose the correct percent rate of change for each function.

Function        25% Growth        25% Decay        75% Growth        75% Decay

ƒ(x) = 0.25x
ƒ(x) = (1 – 0.25)x
ƒ(x) = 1.25x
ƒ(x) = (1 + 0.25)x
ƒ(x) = 0.75x

We are tasked with determining whether the given functions represent growth or decay, and then calculating their percent rate of change. Here's how to analyze each one:

1. **\( f(x) = 0.25^x \)**:
   - The base of the exponential function is \( 0.25 \), which is less than 1.
   - When the base is less than 1, this represents decay.
   - Since \( 0.25 = 1 - 0.75 \), the percent decay is **75%**.
   - **Answer: 75% Decay**

2. **\( f(x) = (1 - 0.25)^x = 0.75^x \)**:
   - The base is \( 0.75 \), which is less than 1.
   - This represents decay.
   - Since \( 1 - 0.25 = 0.75 \), the percent decay is **25%**.
   - **Answer: 25% Decay**

3. **\( f(x) = 1.25^x \)**:
   - The base of the exponential function is \( 1.25 \), which is greater than 1.
   - This represents growth.
   - Since \( 1.25 = 1 + 0.25 \), the percent growth is **25%**.
   - **Answer: 25% Growth**

4. **\( f(x) = (1 + 0.25)^x = 1.25^x \)**:
   - This is identical to the previous function.
   - The base is \( 1.25 \), so this represents growth.
   - The percent growth is **25%**.
   - **Answer: 25% Growth**

5. **\( f(x) = 0.75^x \)**:
   - The base is \( 0.75 \), which is less than 1.
   - This represents decay.
   - Since \( 0.75 = 1 - 0.25 \), the percent decay is **25%**.
   - **Answer: 25% Decay**

### Summary Table:
| Function              | 25% Growth | 25% Decay | 75% Growth | 75% Decay |
|-----------------------|------------|-----------|------------|-----------|
| \( f(x) = 0.25^x \)   |            |           |            |     ✓     |
| \( f(x) = (1 - 0.25)^x = 0.75^x \) |            |     ✓     |            |           |
| \( f(x) = 1.25^x \)   |     ✓      |           |            |           |
| \( f(x) = (1 + 0.25)^x = 1.25^x \) |     ✓      |           |            |           |
| \( f(x) = 0.75^x \)   |            |     ✓     |            |           |

Would you like a more detailed explanation or further clarification?

---

Here are 5 related questions to expand your understanding:
1. What is the general formula for calculating percent change in exponential growth or decay?
2. How do you differentiate between linear and exponential growth?
3. What is the effect of changing the base of an exponential function to a value greater than 2?
4. How do compound interest formulas relate to exponential growth?
5. Can you explain how exponential decay is applied in real-world scenarios, like radioactive decay?

**Tip**: The base of an exponential function determines whether it represents growth (base > 1) or decay (base < 1).

Complete the table to determine the percent rate of change of each function. Choose the correct percent rate of change for each function.

Function        25% Growth        25% Decay        75% Growth        75% Decay

ƒ(x) = 0.25x
ƒ(x) = (1 – 0.25)x
ƒ(x) = 1.25x
ƒ(x) = (1 + 0.25)x
ƒ(x) = 0.75x

Learn how to calculate the percent rate of change in exponential growth and decay functions. This guide covers functions like ƒ(x) = 0.25^x, ƒ(x) = (1 – 0.25)^x, ƒ(x) = 1.25^x, and others. Suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation