The Table Below Shows A Relationship Between Two Variables.$\[ \begin{tabular}{|c|c|} \hline $x$ & $y$ \\ \hline 0 & 200 \\ \hline 1 & 100 \\ \hline 2 & 50 \\ \hline 3 & 25 \\ \hline 4 & 12.5 \\ \hline \end{tabular} \\]Which Equation Best
Introduction
In mathematics, relationships between variables are often represented using equations. These equations can be used to model real-world situations, make predictions, and solve problems. In this article, we will explore a table that shows a relationship between two variables, x and y, and determine which equation best represents this relationship.
The Table
The table below shows the relationship between x and y.
| x | y |
|---|---|
| 0 | 200 |
| 1 | 100 |
| 2 | 50 |
| 3 | 25 |
| 4 | 12.5 |
Analyzing the Relationship
At first glance, the relationship between x and y appears to be decreasing. As x increases, y decreases. However, the rate of decrease is not constant. To determine which equation best represents this relationship, we need to analyze the data further.
Linear Equation
A linear equation is a simple equation that represents a straight line. The general form of a linear equation is:
y = mx + b
where m is the slope and b is the y-intercept.
To determine if the relationship between x and y is linear, we can calculate the slope and y-intercept using the data from the table.
| x | y |
|---|---|
| 0 | 200 |
| 1 | 100 |
| 2 | 50 |
| 3 | 25 |
| 4 | 12.5 |
Using the data from the table, we can calculate the slope and y-intercept as follows:
m = (y2 - y1) / (x2 - x1) = (100 - 200) / (1 - 0) = -100
b = y1 - mx1 = 200 - (-100)(0) = 200
The equation of the line is:
y = -100x + 200
However, this equation does not accurately represent the relationship between x and y. The data from the table shows that the relationship is not linear, but rather exponential.
Exponential Equation
An exponential equation is a type of equation that represents a curve. The general form of an exponential equation is:
y = ab^x
where a is the initial value and b is the growth factor.
To determine if the relationship between x and y is exponential, we can analyze the data from the table.
| x | y |
|---|---|
| 0 | 200 |
| 1 | 100 |
| 2 | 50 |
| 3 | 25 |
| 4 | 12.5 |
The data from the table shows that the relationship between x and y is decreasing exponentially. The initial value is 200, and the growth factor is 0.5.
The equation of the exponential curve is:
y = 200(0.5)^x
This equation accurately represents the relationship between x and y.
Conclusion
In conclusion, the table shows a relationship between two variables, x and y. The data from the table shows that the relationship is not linear, but rather exponential. The equation of the exponential curve is:
y = 200(0.5)^x
This equation accurately represents the relationship between x and y.
References
- [1] Khan Academy. (n.d.). Linear Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f6
- [2] Khan Academy. (n.d.). Exponential Equations. Retrieved from https://www.khanacademy.org/math/algebra/x2f1f7
Frequently Asked Questions
- Q: What is the relationship between x and y? A: The relationship between x and y is exponential.
- Q: What is the equation of the exponential curve? A: The equation of the exponential curve is y = 200(0.5)^x.
- Q: Why is the linear equation not accurate?
A: The linear equation is not accurate because the relationship between x and y is not linear, but rather exponential.
Frequently Asked Questions: Understanding the Relationship Between x and y ====================================================================
Q: What is the relationship between x and y?
A: The relationship between x and y is exponential. As x increases, y decreases, but the rate of decrease is not constant.
Q: What is the equation of the exponential curve?
A: The equation of the exponential curve is y = 200(0.5)^x. This equation accurately represents the relationship between x and y.
Q: Why is the linear equation not accurate?
A: The linear equation is not accurate because the relationship between x and y is not linear, but rather exponential. The linear equation y = -100x + 200 does not accurately represent the data from the table.
Q: How can I determine if the relationship between x and y is linear or exponential?
A: To determine if the relationship between x and y is linear or exponential, you can analyze the data from the table. If the data points are close to a straight line, the relationship is likely linear. If the data points are not close to a straight line, the relationship is likely exponential.
Q: What is the significance of the growth factor in the exponential equation?
A: The growth factor in the exponential equation is 0.5. This means that for every increase in x by 1, y decreases by a factor of 0.5.
Q: How can I use the exponential equation to make predictions?
A: To make predictions using the exponential equation, you can plug in a value for x and solve for y. For example, if you want to know the value of y when x = 5, you can plug in x = 5 into the equation y = 200(0.5)^x and solve for y.
Q: What are some real-world applications of exponential equations?
A: Exponential equations have many real-world applications, including:
- Modeling population growth
- Modeling chemical reactions
- Modeling financial investments
- Modeling the spread of diseases
Q: How can I graph the exponential equation?
A: To graph the exponential equation, you can use a graphing calculator or a computer program. You can also use a table of values to create a graph.
Q: What are some common mistakes to avoid when working with exponential equations?
A: Some common mistakes to avoid when working with exponential equations include:
- Not checking the domain of the equation
- Not checking the range of the equation
- Not using the correct order of operations
- Not simplifying the equation
Q: How can I simplify an exponential equation?
A: To simplify an exponential equation, you can use the following steps:
- Factor out any common factors
- Use the properties of exponents to simplify the equation
- Combine like terms
Q: What are some tips for working with exponential equations?
A: Some tips for working with exponential equations include:
- Use a calculator or computer program to check your work
- Check the domain and range of the equation
- Use the properties of exponents to simplify the equation
- Combine like terms
Conclusion
In conclusion, the relationship between x and y is exponential, and the equation of the exponential curve is y = 200(0.5)^x. This equation accurately represents the relationship between x and y. By following the tips and avoiding common mistakes, you can work with exponential equations with confidence.