{ \begin{array}{|c|c|} \hline x & Y \\ \hline -2 & -5 \\ \hline 2 & 5 \\ \hline 4 & 10 \\ \hline 6 & 15 \\ \hline \end{array} \}$Which Best Describes The Function Represented By The Table?A. Direct Variation; ${$k=\frac{2}{5}\$}$B.
Introduction
Direct variation is a fundamental concept in mathematics that describes the relationship between two variables. It is a type of function where the output value is directly proportional to the input value. In this article, we will explore the concept of direct variation, its applications, and how it can be represented using a table.
What is Direct Variation?
Direct variation is a relationship between two variables, x and y, where the output value (y) is directly proportional to the input value (x). This means that as the input value increases or decreases, the output value also increases or decreases in a consistent manner. The relationship can be represented by the equation y = kx, where k is the constant of proportionality.
Representing Direct Variation Using a Table
A table can be used to represent a direct variation relationship between two variables. The table consists of input values (x) and corresponding output values (y). In the given table, we have the following values:
| x | y |
|---|---|
| -2 | -5 |
| 2 | 5 |
| 4 | 10 |
| 6 | 15 |
Analyzing the Table
Looking at the table, we can see that as the input value (x) increases, the output value (y) also increases in a consistent manner. This suggests that the relationship between x and y is a direct variation. To confirm this, we can calculate the constant of proportionality (k) using the given values.
Calculating the Constant of Proportionality (k)
To calculate the constant of proportionality (k), we can use the following formula:
k = (y2 - y1) / (x2 - x1)
Using the given values, we can calculate k as follows:
k = (5 - (-5)) / (2 - (-2)) k = 10 / 4 k = 2.5
However, we can also calculate k using the ratio of the output values to the input values. Since the output values are directly proportional to the input values, we can calculate k as follows:
k = y / x k = 5 / 2 k = 2.5
Conclusion
Based on the analysis of the table and the calculation of the constant of proportionality (k), we can conclude that the function represented by the table is a direct variation. The constant of proportionality (k) is 2.5, which means that for every unit increase in the input value (x), the output value (y) increases by 2.5 units.
Answer
The correct answer is:
A. Direct variation; k = 2.5
Discussion
Direct variation is a fundamental concept in mathematics that has numerous applications in real-life situations. It is used to model relationships between variables in fields such as physics, engineering, economics, and more. Understanding direct variation and its applications can help us make informed decisions and solve complex problems.
Real-Life Applications of Direct Variation
Direct variation has numerous real-life applications, including:
- Physics: The relationship between the distance traveled by an object and the time taken to travel that distance is a direct variation.
- Engineering: The relationship between the force applied to an object and the resulting displacement is a direct variation.
- Economics: The relationship between the price of a commodity and the quantity demanded is a direct variation.
- Biology: The relationship between the concentration of a substance and its effect on an organism is a direct variation.
Conclusion
Frequently Asked Questions About Direct Variation
Direct variation is a fundamental concept in mathematics that describes the relationship between two variables. In this article, we will answer some frequently asked questions about direct variation.
Q: What is direct variation?
A: Direct variation is a relationship between two variables, x and y, where the output value (y) is directly proportional to the input value (x). This means that as the input value increases or decreases, the output value also increases or decreases in a consistent manner.
Q: How is direct variation represented mathematically?
A: Direct variation is represented mathematically by the equation y = kx, where k is the constant of proportionality.
Q: What is the constant of proportionality (k)?
A: The constant of proportionality (k) is a value that represents the ratio of the output value (y) to the input value (x). It is a key concept in direct variation and can be calculated using various methods.
Q: How do I calculate the constant of proportionality (k)?
A: There are several methods to calculate the constant of proportionality (k). One common method is to use the formula:
k = (y2 - y1) / (x2 - x1)
Another method is to use the ratio of the output values to the input values:
k = y / x
Q: What are some real-life applications of direct variation?
A: Direct variation has numerous real-life applications, including:
- Physics: The relationship between the distance traveled by an object and the time taken to travel that distance is a direct variation.
- Engineering: The relationship between the force applied to an object and the resulting displacement is a direct variation.
- Economics: The relationship between the price of a commodity and the quantity demanded is a direct variation.
- Biology: The relationship between the concentration of a substance and its effect on an organism is a direct variation.
Q: How do I determine if a relationship is a direct variation?
A: To determine if a relationship is a direct variation, you can use the following steps:
- Plot the data points on a graph.
- Check if the graph is a straight line.
- If the graph is a straight line, then the relationship is a direct variation.
Q: What are some common mistakes to avoid when working with direct variation?
A: Some common mistakes to avoid when working with direct variation include:
- Not checking for proportionality: Make sure to check if the relationship is a direct variation by plotting the data points on a graph and checking if the graph is a straight line.
- Not calculating the constant of proportionality (k) correctly: Make sure to calculate the constant of proportionality (k) using the correct formula.
- Not considering the units: Make sure to consider the units of the variables when working with direct variation.
Conclusion
In conclusion, direct variation is a fundamental concept in mathematics that describes the relationship between two variables. It is a type of function where the output value is directly proportional to the input value. Understanding direct variation and its applications can help us make informed decisions and solve complex problems in various fields.