Consider The Two Exponential Equations Shown. Identify The Attributes For Each Equation To Complete The Table.Responses: Decay, 11%, Growth, 40, 89%, 8.9%, 111%, 250$[ \begin{array}{|c|c|} \hline 40=250(0.89)^x & 250=40(1.11)^x \ \hline
**Understanding Exponential Equations: A Comprehensive Guide** ===========================================================
What are Exponential Equations?
Exponential equations are mathematical expressions that involve an exponential function, which is a function that is raised to a power. These equations are used to model real-world situations where a quantity changes at a constant rate over time. In this article, we will explore two exponential equations and identify the attributes for each equation to complete a table.
The Two Exponential Equations
The two exponential equations are:
- 40 = 250(0.89)^x
- 250 = 40(1.11)^x
Attributes of Exponential Equations
To complete the table, we need to identify the attributes of each equation. The attributes include:
- Type of Equation: Is the equation a growth or decay equation?
- Initial Value: What is the initial value of the equation?
- Rate of Change: What is the rate of change of the equation?
- Final Value: What is the final value of the equation?
Completing the Table
Let's complete the table by identifying the attributes of each equation.
| Equation | Type of Equation | Initial Value | Rate of Change | Final Value |
|---|---|---|---|---|
| 40 = 250(0.89)^x | Decay | 250 | 0.89 | 40 |
| 250 = 40(1.11)^x | Growth | 40 | 1.11 | 250 |
Q&A
Q: What is the difference between a growth and decay equation? A: A growth equation is an exponential equation where the value increases over time, while a decay equation is an exponential equation where the value decreases over time.
Q: How do you determine the rate of change of an exponential equation? A: The rate of change of an exponential equation is determined by the base of the exponential function. In the equation 40 = 250(0.89)^x, the rate of change is 0.89, which means that the value decreases by 11% each time x increases by 1.
Q: How do you determine the final value of an exponential equation? A: The final value of an exponential equation is determined by the initial value and the rate of change. In the equation 250 = 40(1.11)^x, the final value is 250, which is 6 times the initial value of 40.
Q: What is the significance of exponential equations in real-world situations? A: Exponential equations are used to model real-world situations where a quantity changes at a constant rate over time. Examples include population growth, chemical reactions, and financial investments.
Q: How do you solve exponential equations? A: Exponential equations can be solved using logarithms. To solve the equation 40 = 250(0.89)^x, we can take the logarithm of both sides and use the properties of logarithms to isolate x.
Conclusion
In conclusion, exponential equations are mathematical expressions that involve an exponential function. The two exponential equations 40 = 250(0.89)^x and 250 = 40(1.11)^x have different attributes, including type of equation, initial value, rate of change, and final value. Understanding these attributes is crucial in solving exponential equations and applying them to real-world situations.
Frequently Asked Questions
- What is the difference between a growth and decay equation?
- A growth equation is an exponential equation where the value increases over time, while a decay equation is an exponential equation where the value decreases over time.
- How do you determine the rate of change of an exponential equation?
- The rate of change of an exponential equation is determined by the base of the exponential function.
- How do you determine the final value of an exponential equation?
- The final value of an exponential equation is determined by the initial value and the rate of change.
- What is the significance of exponential equations in real-world situations?
- Exponential equations are used to model real-world situations where a quantity changes at a constant rate over time.
- How do you solve exponential equations?
- Exponential equations can be solved using logarithms.
Glossary
- Exponential Function: A function that is raised to a power.
- Growth Equation: An exponential equation where the value increases over time.
- Decay Equation: An exponential equation where the value decreases over time.
- Rate of Change: The base of the exponential function.
- Final Value: The value of the equation after a certain period of time.
References
- [1] Khan Academy. (n.d.). Exponential Functions. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f0f/exponential-functions
- [2] Mathway. (n.d.). Exponential Equations. Retrieved from https://www.mathway.com/subjects/exponential-equations
About the Author
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