Determine whether the following equation represents exponential growth or decay. Explain how you know using a complete sentence. ƒ(x) = −2(6) ²- Paragraph BI Uv Αγ lili v !!!! 5" ...
Determine whether the following equation represents exponential growth or decay. Explain how you know using a complete sentence. ƒ(x) = −2(6) ²- Paragraph BI Uv Αγ lili v !!!! 5" ...
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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Question
![### Exponential Growth or Decay
**Problem Statement:**
Determine whether the following equation represents exponential growth or decay. Explain how you know using a complete sentence.
\[ f(x) = -2(6)^{x-3} \]
**Explanation:**
To identify whether the function represents exponential growth or decay, we need to examine the base of the exponential expression and how the exponent affects it.
1. **Base of the Exponential Function:**
- The base of the exponential expression is \(6\), which is greater than 1. Typically, a base greater than 1 indicates an exponential growth scenario.
2. **Exponent Analysis:**
- The exponent is \(x - 3\). As \(x\) increases, \(x - 3\) also increases. Therefore, the term \(6^{x-3}\) increases as \(x\) increases.
3. **Negative Coefficient:**
- The coefficient of the exponential term is \(-2\). This negative coefficient reflects the exponential growth over the x-axis, which might not directly influence the determination of growth or decay.
**Conclusion:**
Despite the negative coefficient of \(-2\), the base of the exponential term \(6\) (which is greater than 1) suggests this function represents exponential growth. The negative sign affects the y-values of the function, reflecting across the x-axis, but doesn't affect the determination of exponential growth or decay.
In a complete sentence:
The equation \( f(x) = -2(6)^{x-3} \) represents exponential growth because the base of the exponential function, 6, is greater than 1.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc3692e55-09d2-4c7f-90e9-0ed90d8b0dce%2F2ceceeb6-3430-45a6-a4dd-2ad1cdb85a2f%2F9pi9los_processed.png&w=3840&q=75)
Transcribed Image Text:### Exponential Growth or Decay
**Problem Statement:**
Determine whether the following equation represents exponential growth or decay. Explain how you know using a complete sentence.
\[ f(x) = -2(6)^{x-3} \]
**Explanation:**
To identify whether the function represents exponential growth or decay, we need to examine the base of the exponential expression and how the exponent affects it.
1. **Base of the Exponential Function:**
- The base of the exponential expression is \(6\), which is greater than 1. Typically, a base greater than 1 indicates an exponential growth scenario.
2. **Exponent Analysis:**
- The exponent is \(x - 3\). As \(x\) increases, \(x - 3\) also increases. Therefore, the term \(6^{x-3}\) increases as \(x\) increases.
3. **Negative Coefficient:**
- The coefficient of the exponential term is \(-2\). This negative coefficient reflects the exponential growth over the x-axis, which might not directly influence the determination of growth or decay.
**Conclusion:**
Despite the negative coefficient of \(-2\), the base of the exponential term \(6\) (which is greater than 1) suggests this function represents exponential growth. The negative sign affects the y-values of the function, reflecting across the x-axis, but doesn't affect the determination of exponential growth or decay.
In a complete sentence:
The equation \( f(x) = -2(6)^{x-3} \) represents exponential growth because the base of the exponential function, 6, is greater than 1.
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