Solutions to the Additional Examples

Oct 13, 2025 by windailys.com 111 views

Introduction

In mathematics, a direct variation function is a type of function that describes a linear relationship between two variables. This type of function is characterized by a constant ratio between the variables, which remains the same for all values of the variables. In this article, we will explore the concept of direct variation functions and how to determine the equation of a direct variation function given two points.

What is a Direct Variation Function?

A direct variation function is a function that can be written in the form y = kx, where k is a constant and x is the independent variable. The constant k is called the slope of the function, and it represents the rate of change of the function with respect to the independent variable. In other words, the slope of a direct variation function tells us how much the dependent variable changes when the independent variable changes by one unit.

Example: Finding the Equation of a Direct Variation Function

Suppose we are given two points on a direct variation function: (-8, -6) and (12, 9). We want to find the equation of the function that passes through these two points. To do this, we need to find the slope of the function, which is represented by the constant k.

Step 1: Find the Slope of the Function

To find the slope of the function, we can use the formula:

k = (y2 - y1) / (x2 - x1)

where (x1, y1) and (x2, y2) are the two given points. Plugging in the values of the two points, we get:

k = (9 - (-6)) / (12 - (-8)) k = (9 + 6) / (12 + 8) k = 15 / 20 k = 3/4

Step 2: Write the Equation of the Function

Now that we have found the slope of the function, we can write the equation of the function in the form y = kx. Plugging in the value of k, we get:

y = (3/4)x

Conclusion

In this article, we have explored the concept of direct variation functions and how to determine the equation of a direct variation function given two points. We have seen that a direct variation function is a type of function that describes a linear relationship between two variables, and that the equation of a direct variation function can be written in the form y = kx, where k is a constant and x is the independent variable. We have also seen how to find the slope of a direct variation function using the formula k = (y2 - y1) / (x2 - x1), and how to write the equation of a direct variation function using the value of k.

Answer to the Problem

Based on the calculations above, the equation of the direct variation function that passes through the points (-8, -6) and (12, 9) is:

y = (3/4)x

This is option C in the problem statement.

Additional Examples

Here are a few more examples of direct variation functions:

Example 1: Find the equation of the direct variation function that passes through the points (2, 3) and (4, 6).
Example 2: Find the equation of the direct variation function that passes through the points (-2, 4) and (3, 9).
Example 3: Find the equation of the direct variation function that passes through the points (1, 2) and (5, 10).

Solutions to the Additional Examples

Example 1: To find the equation of the direct variation function that passes through the points (2, 3) and (4, 6), we can use the formula k = (y2 - y1) / (x2 - x1). Plugging in the values of the two points, we get:

k = (6 - 3) / (4 - 2) k = 3 / 2

The equation of the direct variation function is:

y = (3/2)x

Example 2: To find the equation of the direct variation function that passes through the points (-2, 4) and (3, 9), we can use the formula k = (y2 - y1) / (x2 - x1). Plugging in the values of the two points, we get:

k = (9 - 4) / (3 - (-2)) k = 5 / 5 k = 1

The equation of the direct variation function is:

y = x

Example 3: To find the equation of the direct variation function that passes through the points (1, 2) and (5, 10), we can use the formula k = (y2 - y1) / (x2 - x1). Plugging in the values of the two points, we get:

k = (10 - 2) / (5 - 1) k = 8 / 4 k = 2

The equation of the direct variation function is:

y = 2x

Conclusion

Introduction

In our previous article, we explored the concept of direct variation functions and how to determine the equation of a direct variation function given two points. 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 function that describes a linear relationship between two variables. This type of function is characterized by a constant ratio between the variables, which remains the same for all values of the variables.

Q: How do I determine the equation of a direct variation function given two points?

A: To determine the equation of a direct variation function given two points, you can use the formula k = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two given points. Once you have found the value of k, you can write the equation of the function in the form y = kx.

Q: What is the slope of a direct variation function?

A: The slope of a direct variation function is represented by the constant k. The slope tells us how much the dependent variable changes when the independent variable changes by one unit.

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 formula k = (y2 - y1) / (x2 - x1), where (x1, y1) and (x2, y2) are the two given points.

Q: What is the equation of a direct variation function?

A: The equation of a direct variation function is written in the form y = kx, where k is a constant and x is the independent variable.

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 find the value of k using the formula k = (y2 - y1) / (x2 - x1), and then plug this value into the equation y = kx.

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 can use a graphing calculator or a graphing software. You can also use a coordinate plane to plot the points and draw a line through them.

Q: What are some real-world applications of direct variation functions?

A: Some real-world applications of direct variation functions include:

Physics: The motion of an object under constant acceleration is a direct variation function.
Economics: The demand for a product is a direct variation function of the price of the product.
Engineering: The stress on a material is a direct variation function of the force applied to it.

Conclusion

In this article, we have answered some frequently asked questions about direct variation functions. We have seen that a direct variation function is a type of function that describes a linear relationship between two variables, and that the equation of a direct variation function can be written in the form y = kx, where k is a constant and x is the independent variable. We have also seen how to find the slope of a direct variation function using the formula k = (y2 - y1) / (x2 - x1), and how to write the equation of a direct variation function using the value of k.

Additional Resources

For more information on direct variation functions, you can check out the following resources:

Khan Academy: Direct Variation Functions
Mathway: Direct Variation Functions
Wolfram Alpha: Direct Variation Functions

Practice Problems

Here are some practice problems to help you understand direct variation functions:

Find the equation of the direct variation function that passes through the points (2, 3) and (4, 6).
Find the equation of the direct variation function that passes through the points (-2, 4) and (3, 9).
Find the equation of the direct variation function that passes through the points (1, 2) and (5, 10).

Solutions to the Practice Problems

Find the equation of the direct variation function that passes through the points (2, 3) and (4, 6).

k = (6 - 3) / (4 - 2) k = 3 / 2

The equation of the direct variation function is:

y = (3/2)x

Find the equation of the direct variation function that passes through the points (-2, 4) and (3, 9).

k = (9 - 4) / (3 - (-2)) k = 5 / 5 k = 1

The equation of the direct variation function is:

y = x

Find the equation of the direct variation function that passes through the points (1, 2) and (5, 10).

k = (10 - 2) / (5 - 1) k = 8 / 4 k = 2

The equation of the direct variation function is:

y = 2x