Graphing a Proportional Relationship on the Coordinate Plane

Oct 13, 2025 by magicalceylontours.com 96 views

Introduction

In mathematics, a proportional relationship between two variables, x and y, is a relationship where the ratio of y to x is constant. This means that for every value of x, there is a corresponding value of y that maintains the same ratio. On the coordinate plane, a line can be used to represent a proportional relationship between x and y. In this article, we will explore which point must be on the line that represents a proportional relationship between x and y.

What is a Proportional Relationship?

A proportional relationship is a relationship where the ratio of y to x is constant. This means that for every value of x, there is a corresponding value of y that maintains the same ratio. For example, if the ratio of y to x is 2:1, then for every value of x, the corresponding value of y will be twice that value. This relationship can be represented by the equation y = 2x, where y is proportional to x.

Graphing a Proportional Relationship on the Coordinate Plane

When graphing a proportional relationship on the coordinate plane, we can use the equation y = kx, where k is the constant of proportionality. The graph of this equation will be a straight line that passes through the origin (0, 0). This is because the equation y = kx represents a relationship where the ratio of y to x is constant, and the origin is the point where x = 0 and y = 0.

Which Point Must be on the Line?

Given that the line represents a proportional relationship between x and y, we need to determine which point must be on the line. To do this, we can use the equation y = kx and substitute x = 0 and y = 0 into the equation. This will give us the point that must be on the line.

Substituting x = 0 and y = 0 into the Equation

Substituting x = 0 and y = 0 into the equation y = kx, we get:

y = k(0) y = 0

This shows that when x = 0, y = 0. Therefore, the point (0, 0) must be on the line that represents a proportional relationship between x and y.

Conclusion

In conclusion, the point (0, 0) must be on the line that represents a proportional relationship between x and y. This is because the equation y = kx represents a relationship where the ratio of y to x is constant, and the origin (0, 0) is the point where x = 0 and y = 0.

Answer

The correct answer is D (0, 0).

Discussion

This problem requires an understanding of proportional relationships and how they are represented on the coordinate plane. The equation y = kx represents a proportional relationship between x and y, and the origin (0, 0) is the point where x = 0 and y = 0. Therefore, the point (0, 0) must be on the line that represents a proportional relationship between x and y.

Additional Resources

For more information on proportional relationships and how they are represented on the coordinate plane, see the following resources:

Khan Academy: Proportional Relationships
Math Open Reference: Proportional Relationships
Wolfram MathWorld: Proportional Relationships

References

National Council of Teachers of Mathematics. (2013). Principles to Actions: Ensuring Mathematical Success for All.
Common Core State Standards Initiative. (2010). Common Core State Standards for Mathematics.

Keywords

Proportional relationships
Coordinate plane
Graphing
Equation
Constant of proportionality
Origin
Point on the line
Proportional Relationships Q&A ================================

Frequently Asked Questions

Q: What is a proportional relationship?

A: A proportional relationship is a relationship where the ratio of y to x is constant. This means that for every value of x, there is a corresponding value of y that maintains the same ratio.

Q: How is a proportional relationship represented on the coordinate plane?

A: A proportional relationship is represented on the coordinate plane by a straight line that passes through the origin (0, 0). The equation of this line is y = kx, where k is the constant of proportionality.

Q: What is the constant of proportionality?

A: The constant of proportionality is a number that represents the ratio of y to x in a proportional relationship. It is denoted by the letter k in the equation y = kx.

Q: How do I determine the constant of proportionality?

A: To determine the constant of proportionality, you can use a table of values or a graph to find the ratio of y to x. For example, if the table of values shows that when x = 2, y = 4, then the constant of proportionality is 2.

Q: What is the origin in a proportional relationship?

A: The origin is the point where x = 0 and y = 0. In a proportional relationship, the origin is the point where the line passes through.

Q: How do I graph a proportional relationship?

A: To graph a proportional relationship, you can use the equation y = kx and substitute different values of x to find the corresponding values of y. Then, plot the points on the coordinate plane and draw a straight line through them.

Q: What is the equation of a proportional relationship?

A: The equation of a proportional relationship is y = kx, where k is the constant of proportionality.

Q: How do I determine if a relationship is proportional?

A: To determine if a relationship is proportional, you can use the following steps:

Create a table of values or a graph of the relationship.
Check if the ratio of y to x is constant.
If the ratio is constant, then the relationship is proportional.

Q: What are some real-world examples of proportional relationships?

A: Some real-world examples of proportional relationships include:

The cost of a product and the number of units sold
The distance traveled and the time taken
The amount of money earned and the number of hours worked

Q: How do I solve problems involving proportional relationships?

A: To solve problems involving proportional relationships, you can use the following steps:

Read the problem carefully and identify the variables.
Create a table of values or a graph of the relationship.
Use the equation y = kx to find the constant of proportionality.
Substitute the values of x and y into the equation to find the solution.

Q: What are some common mistakes to avoid when working with proportional relationships?

A: Some common mistakes to avoid when working with proportional relationships include:

Assuming that a relationship is proportional when it is not.
Failing to check if the ratio of y to x is constant.
Not using the correct equation to represent the relationship.

Q: How do I check if a relationship is proportional?

A: To check if a relationship is proportional, you can use the following steps:

Create a table of values or a graph of the relationship.
Check if the ratio of y to x is constant.
If the ratio is constant, then the relationship is proportional.

Q: What are some tips for working with proportional relationships?

A: Some tips for working with proportional relationships include:

Always check if the ratio of y to x is constant.
Use the correct equation to represent the relationship.
Create a table of values or a graph to visualize the relationship.
Use real-world examples to illustrate the concept of proportional relationships.