Determine Whether The Function Represents Exponential Growth Or Decay, Then Identify The Correct Graph Of The Function.$\[ Y = 2 \cdot 2^{x-1} - 2 \\]Choose 2:A. Exponential GrowthB. Exponential Decay

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Understanding Exponential Functions

Exponential functions are a type of mathematical function that describes a relationship between two variables, where one variable is a constant power of the other variable. These functions are commonly used to model real-world phenomena, such as population growth, chemical reactions, and financial investments. In this article, we will explore how to determine whether a given function represents exponential growth or decay and identify the correct graph of the function.

Exponential Growth vs. Exponential Decay

Exponential growth and decay are two types of exponential functions that have different characteristics. Exponential growth occurs when a quantity increases at a rate proportional to its current value, resulting in a rapid increase in the quantity over time. On the other hand, exponential decay occurs when a quantity decreases at a rate proportional to its current value, resulting in a rapid decrease in the quantity over time.

Identifying Exponential Growth or Decay

To determine whether a function represents exponential growth or decay, we need to examine the coefficient of the exponential term. If the coefficient is positive, the function represents exponential growth. If the coefficient is negative, the function represents exponential decay.

Analyzing the Given Function

The given function is:

y=2⋅2x−1−2{ y = 2 \cdot 2^{x-1} - 2 }

To determine whether this function represents exponential growth or decay, we need to examine the coefficient of the exponential term. In this case, the coefficient is 2, which is positive. Therefore, the function represents exponential growth.

Simplifying the Function

To simplify the function, we can rewrite it as:

y=2⋅2x−1−2{ y = 2 \cdot 2^{x-1} - 2 } y=2x⋅2−1−2{ y = 2^{x} \cdot 2^{-1} - 2 } y=12⋅2x−2{ y = \frac{1}{2} \cdot 2^{x} - 2 }

This simplified form of the function makes it easier to analyze and understand its behavior.

Graphing the Function

To graph the function, we can use a graphing calculator or software. The graph of the function will be a curve that increases rapidly as x increases. The graph will also have a horizontal asymptote at y = 2.

Conclusion

In conclusion, the given function represents exponential growth. The coefficient of the exponential term is positive, indicating that the function increases rapidly as x increases. The graph of the function will be a curve that increases rapidly as x increases, with a horizontal asymptote at y = 2.

Key Takeaways

  • Exponential growth occurs when a quantity increases at a rate proportional to its current value.
  • Exponential decay occurs when a quantity decreases at a rate proportional to its current value.
  • The coefficient of the exponential term determines whether a function represents exponential growth or decay.
  • The graph of an exponential function will be a curve that increases or decreases rapidly as x increases.

Final Answer

The final answer is:

A. Exponential growth

Additional Resources

For more information on exponential functions and graphing, please refer to the following resources:

  • Khan Academy: Exponential Functions
  • Mathway: Exponential Functions
  • Wolfram Alpha: Exponential Functions

References

  • Larson, R. (2019). Calculus. Cengage Learning.
  • Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.
  • Anton, H. (2019). Calculus: A New Horizon. John Wiley & Sons.

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Frequently Asked Questions

In this article, we will answer some of the most frequently asked questions about exponential growth and decay.

Q: What is exponential growth?

A: Exponential growth is a type of growth where a quantity increases at a rate proportional to its current value. This means that the rate of growth is not constant, but rather increases as the quantity grows.

Q: What is exponential decay?

A: Exponential decay is a type of decay where a quantity decreases at a rate proportional to its current value. This means that the rate of decay is not constant, but rather increases as the quantity decreases.

Q: How do I determine whether a function represents exponential growth or decay?

A: To determine whether a function represents exponential growth or decay, you need to examine the coefficient of the exponential term. If the coefficient is positive, the function represents exponential growth. If the coefficient is negative, the function represents exponential decay.

Q: What is the difference between exponential growth and linear growth?

A: Exponential growth and linear growth are two different types of growth. Linear growth occurs when a quantity increases at a constant rate, whereas exponential growth occurs when a quantity increases at a rate proportional to its current value.

Q: Can you give an example of exponential growth in real life?

A: Yes, a classic example of exponential growth is population growth. When a population grows exponentially, the number of individuals increases rapidly, leading to a rapid increase in the population size.

Q: Can you give an example of exponential decay in real life?

A: Yes, a classic example of exponential decay is radioactive decay. When a radioactive substance decays exponentially, the amount of the substance decreases rapidly, leading to a rapid decrease in the substance's concentration.

Q: How do I graph an exponential function?

A: To graph an exponential function, you can use a graphing calculator or software. The graph of an exponential function will be a curve that increases or decreases rapidly as x increases.

Q: What is the horizontal asymptote of an exponential function?

A: The horizontal asymptote of an exponential function is the value that the function approaches as x approaches infinity. For an exponential function of the form y = a^x, the horizontal asymptote is y = 0 if a < 1 and y = infinity if a > 1.

Q: Can you give an example of an exponential function?

A: Yes, a classic example of an exponential function is the function y = 2^x. This function represents exponential growth, as the value of y increases rapidly as x increases.

Q: Can you give an example of an exponential decay function?

A: Yes, a classic example of an exponential decay function is the function y = e^(-x). This function represents exponential decay, as the value of y decreases rapidly as x increases.

Conclusion

In conclusion, exponential growth and decay are two important concepts in mathematics that have many real-world applications. By understanding these concepts, you can better analyze and model real-world phenomena.

Key Takeaways

  • Exponential growth occurs when a quantity increases at a rate proportional to its current value.
  • Exponential decay occurs when a quantity decreases at a rate proportional to its current value.
  • The coefficient of the exponential term determines whether a function represents exponential growth or decay.
  • The graph of an exponential function will be a curve that increases or decreases rapidly as x increases.

Final Answer

The final answer is:

A. Exponential growth and decay are two important concepts in mathematics that have many real-world applications.

Additional Resources

For more information on exponential growth and decay, please refer to the following resources:

  • Khan Academy: Exponential Functions
  • Mathway: Exponential Functions
  • Wolfram Alpha: Exponential Functions

References

  • Larson, R. (2019). Calculus. Cengage Learning.
  • Stewart, J. (2019). Calculus: Early Transcendentals. Cengage Learning.
  • Anton, H. (2019). Calculus: A New Horizon. John Wiley & Sons.

License

This article is licensed under the Creative Commons Attribution-ShareAlike 4.0 International License.