A Direct Variation Function Contains The Points { (-9, -3)$}$ And { (-12, -4)$}$. Which Equation Represents The Function?A. { Y = -3x$}$B. { Y = -\frac{x}{3}$}$C. { Y = \frac{x}{3}$} D . \[ D. \[ D . \[ Y =
Direct Variation Functions: Understanding the Relationship Between Variables
In mathematics, a direct variation function is a type of linear function that describes a relationship between two variables, where one variable is a constant multiple of the other. This type of function is often represented by the equation y = kx, where k is the constant of variation. In this article, we will explore the concept of direct variation functions and use the given points to determine the equation that represents the function.
What is a Direct Variation Function?
A direct variation function is a linear function that can be represented by the equation y = kx, where k is the constant of variation. This means that as the value of x increases or decreases, the value of y also increases or decreases at a constant rate. The constant of variation, k, determines the rate at which y changes in response to changes in x.
Example: Using Given Points to Determine the Equation
We are given two points, (-9, -3) and (-12, -4), that lie on the direct variation function. To determine the equation that represents the function, we need to find the constant of variation, k.
Step 1: Find the Constant of Variation
To find the constant of variation, we can use the given points to set up a system of equations. We know that the equation of the direct variation function is y = kx. We can substitute the given points into this equation to get two equations:
-3 = k(-9) -4 = k(-12)
Step 2: Solve for k
We can solve for k by dividing both sides of each equation by the corresponding value of x.
k = -3 / (-9) k = -4 / (-12)
k = 1/3 k = 1/3
Step 3: Write the Equation
Now that we have found the constant of variation, k, we can write the equation that represents the direct variation function. Since k = 1/3, we can substitute this value into the equation y = kx to get:
y = (1/3)x
In this article, we explored the concept of direct variation functions and used the given points to determine the equation that represents the function. We found that the constant of variation, k, is 1/3, and the equation that represents the function is y = (1/3)x. This equation describes a direct variation relationship between the variables x and y, where y is a constant multiple of x.
The correct answer is B. y = -\frac{x}{3}.
Direct Variation Functions: Frequently Asked Questions
In our previous article, we explored the concept of direct variation functions and used the given points to determine the equation that represents the function. In this article, we will answer some frequently asked questions about direct variation functions to help you better understand this topic.
Q: What is the difference between a direct variation function and a linear function?
A: A direct variation function is a type of linear function that describes a relationship between two variables, where one variable is a constant multiple of the other. In other words, a direct variation function is a linear function where the slope is not equal to 1.
Q: How do I determine the equation of a direct variation function?
A: To determine the equation of a direct variation function, you need to find the constant of variation, k. You can do this by using the given points to set up a system of equations and solving for k.
Q: What is the constant of variation, k?
A: The constant of variation, k, is a number that determines the rate at which y changes in response to changes in x. It is a measure of the steepness of the direct variation function.
Q: How do I find the constant of variation, k?
A: To find the constant of variation, k, you can use the given points to set up a system of equations and solve for k. You can also use the formula k = y/x, where y is the value of the function at a given point and x is the corresponding value of the independent variable.
Q: What is the equation of a direct variation function?
A: The equation of a direct variation function is y = kx, where k is the constant of variation.
Q: Can a direct variation function have a negative constant of variation?
A: Yes, a direct variation function can have a negative constant of variation. In this case, the function will have a negative slope.
Q: Can a direct variation function have a zero constant of variation?
A: No, a direct variation function cannot have a zero constant of variation. If the constant of variation is zero, the function will be a horizontal line.
Q: Can a direct variation function have a fractional constant of variation?
A: Yes, a direct variation function can have a fractional constant of variation. In this case, the function will have a fractional slope.
Q: Can a direct variation function have a negative fractional constant of variation?
A: Yes, a direct variation function can have a negative fractional constant of variation. In this case, the function will have a negative fractional slope.
In this article, we answered some frequently asked questions about direct variation functions to help you better understand this topic. We hope that this article has been helpful in clarifying any confusion you may have had about direct variation functions.
If you are looking for additional resources to help you learn more about direct variation functions, we recommend the following:
- Khan Academy: Direct Variation
- Mathway: Direct Variation *fram Alpha: Direct Variation
To help you practice what you have learned about direct variation functions, we recommend the following practice problems:
- Find the equation of a direct variation function that passes through the points (2, 6) and (4, 12).
- Find the constant of variation, k, for the direct variation function y = 2x.
- Find the equation of a direct variation function that has a constant of variation, k, of 3/4.
We hope that these practice problems will help you to better understand direct variation functions and to develop your problem-solving skills.