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. \[$y = 14x\$\]

Mar 1, 2025 by ADMIN 205 views

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 (2,14) and (4,28) 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, the value of y also increases at a constant rate. The constant of variation, k, is a measure of how much y changes when x changes by a certain amount.

Example: Using Given Points to Determine the Equation

Let's use the given points (2,14) and (4,28) to determine the equation that represents the function. To do this, we need to find the constant of variation, k.

We can start by using the first point (2,14) and substituting it into the equation y = kx. This gives us:

14 = k(2)

To solve for k, we can divide both sides of the equation by 2:

k = 14/2 k = 7

Now that we have found the constant of variation, k, we can use it to write the equation that represents the function. Since k = 7, we can substitute this value into the equation y = kx to get:

y = 7x

This is the equation that represents the function.

Why is the Equation y = 7x the Correct Answer?

The equation y = 7x is the correct answer because it satisfies the given points (2,14) and (4,28). To see why, let's substitute the values of x and y from each point into the equation:

For the point (2,14):

14 = 7(2) 14 = 14

This shows that the equation y = 7x is true for the point (2,14).

For the point (4,28):

28 = 7(4) 28 = 28

This shows that the equation y = 7x is also true for the point (4,28).

Comparing the Options

Now that we have found the equation that represents the function, let's compare it to the other options:

A. y = x/14 B. y = x/7 C. y = 7x D. y = 14x

The only option that matches the equation we found is option C, y = 7x.

Conclusion

In this article, we used the given points (2,14) and (4,28) to determine the equation that represents the direct variation function. We found that the equation y = 7x is the correct answer because it satisfies the given points. This equation represents a direct variation function, where y is a constant multiple of x.

Key Takeaways

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.
The equation y = kx represents a direct variation function, where k is the constant of variation.
To determine the equation that represents a direct variation function, we can use given points and substitute them into the equation y = kx.
The constant of variation, k, can be found by dividing the value of y by the value of x.

Frequently Asked Questions

Q: What is a direct variation 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.
Q: How do I determine the equation that represents a direct variation function? A: To determine the equation that represents a direct variation function, we can use given points and substitute them into the equation y = kx.
Q: What is the constant of variation, k? A: The constant of variation, k, is a measure of how much y changes when x changes by a certain amount.

References

[1] "Direct Variation Functions" by Math Open Reference
[2] "Direct Variation" by Khan Academy
[3] "Linear Functions" by Purplemath
Direct Variation Functions: Q&A

In our previous article, we explored the concept of direct variation functions and used the given points (2,14) and (4,28) to determine the equation that represents the function. In this article, we will answer some frequently asked questions about direct variation functions.

Q: What is a direct variation 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. This type of function is often represented by the equation y = kx, where k is the constant of variation.

Q: How do I determine the equation that represents a direct variation function?

A: To determine the equation that represents a direct variation function, you can use given points and substitute them into the equation y = kx. You can also use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the constant of variation, k?

A: The constant of variation, k, is a measure of how much y changes when x changes by a certain amount. It is a key component of a direct variation function and can be found by dividing the value of y by the value of x.

Q: How do I find the constant of variation, k?

A: To find the constant of variation, k, you can use the given points and substitute them into the equation y = kx. You can also use the slope-intercept form of a linear equation, y = mx + b, where m is the slope and b is the y-intercept.

Q: What is the difference between a direct variation function and an inverse variation 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. An inverse variation function, on the other hand, is a type of linear function that describes a relationship between two variables, where one variable is a constant multiple of the reciprocal of the other.

Q: Can a direct variation function have a negative constant of variation, k?

A: Yes, a direct variation function can have a negative constant of variation, k. This means that as x increases, y decreases, and vice versa.

Q: Can a direct variation function have a zero constant of variation, k?

A: No, a direct variation function cannot have a zero constant of variation, k. This is because a zero constant of variation would mean that y is not changing at all, which is not a valid relationship between two variables.

Q: Can a direct variation function have a fractional constant of variation, k?

A: Yes, a direct variation function can have a fractional constant of variation, k. This means that as x increases, y increases at a rate that is a fraction of the increase in x.

Q: Can a direct variation function have a negative fractional constant of variation, k?

A: Yes, a direct variation function can have a negative fractional constant of variation, k. This means that as x increases, y decreases at a rate that is a fraction of the increase in x.

Q: How do I graph a direct variation function?

A: To graph a direct variation function, you can use the equation y = kx and plot the points (0,0) and (1,k) on a coordinate plane. You can then draw a line through these points to represent the function.

Q: How do I find the slope of a direct variation function?

A: To find the slope of a direct variation function, you can use the equation y = kx and divide the value of k by 1. This will give you the slope of the function.

Q: How do I find the y-intercept of a direct variation function?

A: To find the y-intercept of a direct variation function, you can use the equation y = kx and substitute x = 0 into the equation. This will give you the y-intercept of the function.

Q: Can a direct variation function have a horizontal asymptote?

A: No, a direct variation function cannot have a horizontal asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function have a vertical asymptote?

A: No, a direct variation function cannot have a vertical asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function have a slant asymptote?

A: No, a direct variation function cannot have a slant asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a quadratic function?

A: No, a direct variation function cannot be a quadratic function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a polynomial function?

A: No, a direct variation function cannot be a polynomial function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a rational function?

A: No, a direct variation function cannot be a rational function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a trigonometric function?

A: No, a direct variation function cannot be a trigonometric function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be an exponential function?

A: No, a direct variation function cannot be an exponential function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a logarithmic function?

A: No, a direct variation function cannot be a logarithmic function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a piecewise function?

A: No, a direct variation function cannot be a piecewise function. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a domain restriction?

A: No, a direct variation function cannot be a function with a domain restriction. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a range restriction?

A: No, a direct variation function cannot be a function with a range restriction. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a discontinuity?

A: No, a direct variation function cannot be a function with a discontinuity. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a hole?

A: No, a direct variation function cannot be a function with a hole. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a vertical asymptote?

A: No, a direct variation function cannot be a function with a vertical asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a horizontal asymptote?

A: No, a direct variation function cannot be a function with a horizontal asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a slant asymptote?

A: No, a direct variation function cannot be a function with a slant asymptote. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a periodic behavior?

A: No, a direct variation function cannot be a function with a periodic behavior. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

Q: Can a direct variation function be a function with a non-linear behavior?

A: No, a direct variation function cannot be a function with a non-linear behavior. This is because a direct variation function is a linear function that describes a relationship between two variables, where one variable is a constant multiple of the other.

**Q: Can a direct variation function be a function with a non-monotonic behavior