Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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![### Exponential Decay Equations
**Question:**
Select all of the equations that will produce exponential decay.
1. \( y = (0.1)^x \)
2. \( y = 2\left(\frac{1}{4}\right)^x \)
3. \( y = (3)^x \)
4. \( y = \frac{1}{2} (5)^x \)
5. \( y = (2)^x \)
6. \( y = 3(0.4)^x \)
**Options:**
- [ ] \( y = (0.1)^x \)
- [ ] \( y = 2\left(\frac{1}{4}\right)^x \)
- [ ] \( y = (3)^x \)
- [ ] \( y = \frac{1}{2} (5)^x \)
- [ ] \( y = (2)^x \)
- [ ] \( y = 3(0.4)^x \)
---
### Explanation:
In exponential functions, \( y = ab^x \), exponential decay occurs when the base \( b \) (where \( b \) is a positive real number) is between 0 and 1. The following equations represent exponential decay:
1. \( y = (0.1)^x \)
- Here, the base \( 0.1 \) is between 0 and 1.
2. \( y = 2\left(\frac{1}{4}\right)^x \)
- Here, the base \( \frac{1}{4} \) (which is 0.25) is between 0 and 1.
3. \( y = 3(0.4)^x \)
- Here, the base \( 0.4 \) is between 0 and 1.
These functions will decrease as \( x \) increases, which characterizes exponential decay. The other options have bases greater than 1, which do not represent exponential decay.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F202541d0-a01d-4a8f-bf3c-1f74416d2e9a%2Faed214f0-26eb-4b9c-a2b4-7b1471769eda%2F1of9b9k_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exponential Decay Equations
**Question:**
Select all of the equations that will produce exponential decay.
1. \( y = (0.1)^x \)
2. \( y = 2\left(\frac{1}{4}\right)^x \)
3. \( y = (3)^x \)
4. \( y = \frac{1}{2} (5)^x \)
5. \( y = (2)^x \)
6. \( y = 3(0.4)^x \)
**Options:**
- [ ] \( y = (0.1)^x \)
- [ ] \( y = 2\left(\frac{1}{4}\right)^x \)
- [ ] \( y = (3)^x \)
- [ ] \( y = \frac{1}{2} (5)^x \)
- [ ] \( y = (2)^x \)
- [ ] \( y = 3(0.4)^x \)
---
### Explanation:
In exponential functions, \( y = ab^x \), exponential decay occurs when the base \( b \) (where \( b \) is a positive real number) is between 0 and 1. The following equations represent exponential decay:
1. \( y = (0.1)^x \)
- Here, the base \( 0.1 \) is between 0 and 1.
2. \( y = 2\left(\frac{1}{4}\right)^x \)
- Here, the base \( \frac{1}{4} \) (which is 0.25) is between 0 and 1.
3. \( y = 3(0.4)^x \)
- Here, the base \( 0.4 \) is between 0 and 1.
These functions will decrease as \( x \) increases, which characterizes exponential decay. The other options have bases greater than 1, which do not represent exponential decay.
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