A Direct Variation Function Contains The Points \[$(2, 14)\$\] And \[$(4, 28)\$\]. Which Equation Represents The Function?A. \[$y=\frac{x}{14}\$\] B. \[$y=\frac{x}{7}\$\] C. \[$y=7x\$\] D. \[$y=14x\$\]
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Introduction
In mathematics, a direct variation function is a type of function where the output value is directly proportional to the input value. This means that as the input value increases or decreases, the output value also increases or decreases at a constant rate. In this article, we will explore how to find the equation of a direct variation function given two points.
What is a Direct Variation Function?
A direct variation function can be represented by the equation y = kx, where y is the output value, x is the input value, and k is the constant of variation. The constant of variation, k, is a measure of how much the output value changes when the input value changes by one unit.
Finding the Equation of a Direct Variation Function
To find the equation of a direct variation function, we need to use the two given points to determine the constant of variation, k. We can do this by substituting the values of the two points into the equation y = kx and solving for k.
Example
Suppose we are given the two points (2, 14) and (4, 28). We can substitute these values into the equation y = kx to get:
14 = k(2) 28 = k(4)
We can solve for k by dividing both sides of the equation by 2 and 4, respectively:
k = 14/2 k = 28/4 k = 7 k = 7
Since both equations give us the same value for k, we can conclude that the constant of variation is 7.
Writing the Equation of a Direct Variation Function
Now that we have found the constant of variation, k, we can write the equation of the direct variation function. We can do this by substituting the value of k into the equation y = kx:
y = 7x
This is the equation of the direct variation function that passes through the points (2, 14) and (4, 28).
Checking the Answer Choices
Now that we have found the equation of the direct variation function, we can check the answer choices to see which one matches our equation.
- A. y = x/14: This equation does not match our equation, since the constant of variation is 7, not 1/14.
- B. y = x/7: This equation does not match our equation, since the constant of variation is 7, not 1/7.
- C. y = 7x: This equation matches our equation, since the constant of variation is 7.
- D. y = 14x: This equation does not match our equation, since the constant of variation is 7, not 14.
Therefore, the correct answer is C. y = 7x.
Conclusion
In this article, we have explored how to find the equation of a direct variation function given two points. We have seen that the equation of a direct variation function can be represented by the equation y = kx, where y is the output value, x is the input value, and k is the constant of variation. We have also seen how to use the two given points to determine the constant of variation, k, and how to write the equation of the direct variation function. Finally, we have checked the answer choices to see which one matches our equation.
Frequently Asked Questions
Q: What is a direct variation function?
A: A direct variation function is a type of function where the output value is directly proportional to the input value.
Q: How do I find the equation of a direct variation function given two points?
A: To find the equation of a direct variation function, you need to use the two given points to determine the constant of variation, k. You can do this by substituting the values of the two points into the equation y = kx and solving for k.
Q: What is the constant of variation, k?
A: The constant of variation, k, is a measure of how much the output value changes when the input value changes by one unit.
Q: How do I write the equation of a direct variation function?
A: To write the equation of a direct variation function, you need to substitute the value of k into the equation y = kx.
References
Further Reading
Related Topics
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Q: What is a direct variation function?
A: A direct variation function is a type of function where the output value is directly proportional to the input value. This means that as the input value increases or decreases, the output value also increases or decreases at a constant rate.
Q: How do I find the equation of a direct variation function given two points?
A: To find the equation of a direct variation function, you need to use the two given points to determine the constant of variation, k. You can do this by substituting the values of the two points into the equation y = kx and solving for k.
Q: What is the constant of variation, k?
A: The constant of variation, k, is a measure of how much the output value changes when the input value changes by one unit. It is a constant that is used in the equation of a direct variation function to describe the relationship between the input and output values.
Q: How do I write the equation of a direct variation function?
A: To write the equation of a direct variation function, you need to substitute the value of k into the equation y = kx. For example, if the constant of variation is 7, the equation of the direct variation function would be y = 7x.
Q: What are some examples of direct variation functions?
A: Some examples of direct variation functions include:
- y = 2x
- y = 3x
- y = 4x
- y = 5x
Q: How do I graph a direct variation function?
A: To graph a direct variation function, you need to plot the points that satisfy the equation of the function. For example, if the equation of the function is y = 2x, you would plot the points (0, 0), (1, 2), (2, 4), and so on.
Q: What are some real-world applications of direct variation functions?
A: Some real-world applications of direct variation functions include:
- Modeling the relationship between the amount of money spent on a product and the number of units sold
- Modeling the relationship between the amount of time spent on a task and the amount of work completed
- Modeling the relationship between the amount of fuel consumed by a vehicle and the distance traveled
Q: How do I determine if a function is a direct variation function?
A: To determine if a function is a direct variation function, you need to check if the equation of the function is in the form y = kx, where k is a constant. If the equation is in this form, then the function is a direct variation function.
Q: What are some common mistakes to avoid when working with direct variation functions?
A: Some common mistakes to avoid when working with direct variation functions include:
- Confusing direct variation with inverse variation
- Failing to check if the equation of the function is in the form y = kx
- Failing to substitute the correct values into the equation of the function
Q: How do I use direct variation functions to solve real-world problems?
A: To use direct variation functions to solve real-world problems, you need to:
- Identify the variables involved in the problem
- Determine the relationship between the variables
- Write an equation that describes the relationship between the variables
- Solve the equation to find the solution to the problem
Q: What are some resources for learning more about direct variation functions?
A: Some resources for learning more about direct variation functions include:
- Online tutorials and videos
- Textbooks and workbooks
- Online courses and degree programs
- Professional development workshops and conferences
Q: How do I practice working with direct variation functions?
A: To practice working with direct variation functions, you can:
- Work through practice problems and exercises
- Complete online quizzes and assessments
- Participate in online forums and discussion groups
- Join a study group or find a study partner
Q: What are some common applications of direct variation functions in science and engineering?
A: Some common applications of direct variation functions in science and engineering include:
- Modeling the relationship between the amount of fuel consumed by a vehicle and the distance traveled
- Modeling the relationship between the amount of time spent on a task and the amount of work completed
- Modeling the relationship between the amount of money spent on a product and the number of units sold
Q: How do I use direct variation functions to model real-world phenomena?
A: To use direct variation functions to model real-world phenomena, you need to:
- Identify the variables involved in the phenomenon
- Determine the relationship between the variables
- Write an equation that describes the relationship between the variables
- Solve the equation to find the solution to the problem
Q: What are some common challenges when working with direct variation functions?
A: Some common challenges when working with direct variation functions include:
- Confusing direct variation with inverse variation
- Failing to check if the equation of the function is in the form y = kx
- Failing to substitute the correct values into the equation of the function
Q: How do I overcome common challenges when working with direct variation functions?
A: To overcome common challenges when working with direct variation functions, you need to:
- Review the basics of direct variation functions
- Practice working with direct variation functions
- Seek help from a teacher or tutor
- Join a study group or find a study partner