In What Quadrants Of The Coordinate Plane Is The Graph Of The Direct Proportion Located, Which Is Parallel To The Graph Expressed By The Following Formula?$y = 0.8x - 1.6$Answer:The Direct Proportion Is Parallel To The Given Graph. The Graph

Feb 27, 2025 by ADMIN 242 views

**Understanding Direct Proportion and Coordinate Plane**

Introduction

In mathematics, a direct proportion is a relationship between two variables where one variable is a constant multiple of the other. This relationship is often represented graphically on a coordinate plane. The graph of a direct proportion is a straight line that passes through the origin (0, 0) and has a positive slope. In this article, we will explore the quadrants of the coordinate plane where the graph of a direct proportion is located, with a focus on the given formula $y = 0.8x - 1.6$ .

What is a Direct Proportion?

A direct proportion is a relationship between two variables where one variable is a constant multiple of the other. This means that as one variable increases, the other variable also increases at a constant rate. Mathematically, this can be represented as $y = kx$ , where $k$ is a constant and $x$ and $y$ are the variables.

Graph of a Direct Proportion

The graph of a direct proportion is a straight line that passes through the origin (0, 0) and has a positive slope. The slope of the line represents the constant rate at which the variables change. In the case of the given formula $y = 0.8x - 1.6$ , the slope is 0.8, which means that for every unit increase in $x$ , $y$ increases by 0.8 units.

Quadrants of the Coordinate Plane

The coordinate plane is divided into four quadrants, labeled I, II, III, and IV. Each quadrant represents a different region of the plane, with the x-axis and y-axis serving as boundaries.

Quadrant I: This quadrant contains all points where both $x$ and $y$ are positive.
Quadrant II: This quadrant contains all points where $x$ is negative and $y$ is positive.
Quadrant III: This quadrant contains all points where both $x$ and $y$ are negative.
Quadrant IV: This quadrant contains all points where $x$ is positive and $y$ is negative.

Graph of the Given Formula

The given formula $y = 0.8x - 1.6$ represents a direct proportion. To find the graph of this formula, we can start by finding the y-intercept, which is the point where the line intersects the y-axis. In this case, the y-intercept is -1.6.

To find the x-intercept, we can set $y$ equal to 0 and solve for $x$ . This gives us:

0 = 0.8x - 1.6

Solving for $x$ , we get:

x = \frac{1.6}{0.8} = 2

So, the x-intercept is 2.

Finding the Quadrants of the Graph

To find the quadrants of the graph, we can use the x and y intercepts. Since the x-intercept is 2 and the y-intercept is -1.6, the graph passes through the point (2, 0) and the point (0, -1.6).

Since the x-intercept is positive and the y-intercept is negative, the graph passes through Quadrant IV.

Conclusion

In conclusion, the graph of the direct proportion $y = 0.8x - 1.6$ is located in Quadrant IV of the coordinate plane. The graph passes through the origin (0, 0) and has a positive slope, representing a direct proportion. Understanding the quadrants of the coordinate plane is essential in graphing and analyzing mathematical relationships.

References

[1] "Direct Proportion." Math Open Reference, mathopenref.com/direct_proportion.html.
[2] "Coordinate Plane." Math Is Fun, mathisfun.com/geometry/coordinat.html.

Additional Resources

For more information on direct proportion and coordinate plane, visit the following websites:
- Math Open Reference: https://mathopenref.com/
- Math Is Fun: https://mathisfun.com/

Frequently Asked Questions

Q: What is a direct proportion? A: A direct proportion is a relationship between two variables where one variable is a constant multiple of the other.
Q: What is the graph of a direct proportion? A: The graph of a direct proportion is a straight line that passes through the origin (0, 0) and has a positive slope.
Q: What are the quadrants of the coordinate plane? A: The coordinate plane is divided into four quadrants, labeled I, II, III, and IV. Each quadrant represents a different region of the plane.
Direct Proportion and Coordinate Plane: Frequently Asked Questions ====================================================================

Q: What is a direct proportion?

A: A direct proportion is a relationship between two variables where one variable is a constant multiple of the other. This means that as one variable increases, the other variable also increases at a constant rate.

Q: What is the graph of a direct proportion?

A: The graph of a direct proportion is a straight line that passes through the origin (0, 0) and has a positive slope. The slope of the line represents the constant rate at which the variables change.

Q: What are the quadrants of the coordinate plane?

A: The coordinate plane is divided into four quadrants, labeled I, II, III, and IV. Each quadrant represents a different region of the plane.

Quadrant I: This quadrant contains all points where both $x$ and $y$ are positive.
Quadrant II: This quadrant contains all points where $x$ is negative and $y$ is positive.
Quadrant III: This quadrant contains all points where both $x$ and $y$ are negative.
Quadrant IV: This quadrant contains all points where $x$ is positive and $y$ is negative.

Q: How do I find the quadrants of the graph of a direct proportion?

A: To find the quadrants of the graph of a direct proportion, you can use the x and y intercepts. The x-intercept is the point where the line intersects the x-axis, and the y-intercept is the point where the line intersects the y-axis.

Q: What is the significance of the slope in a direct proportion?

A: The slope of a direct proportion represents the constant rate at which the variables change. A positive slope indicates that as one variable increases, the other variable also increases at a constant rate.

Q: Can a direct proportion have a negative slope?

A: No, a direct proportion cannot have a negative slope. A negative slope would indicate that as one variable increases, the other variable decreases, which is not a direct proportion.

Q: How do I graph a direct proportion?

A: To graph a direct proportion, you can start by finding the y-intercept, which is the point where the line intersects the y-axis. Then, you can use the x and y intercepts to find the quadrants of the graph.

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

A: Direct proportion has many real-world applications, including:

Physics: Direct proportion is used to describe the relationship between force and acceleration.
Economics: Direct proportion is used to describe the relationship between supply and demand.
Biology: Direct proportion is used to describe the relationship between population growth and resource availability.

Q: Can I use a calculator to graph a direct proportion?

A: Yes, you can use a calculator to graph a direct proportion. Many graphing calculators have built-in functions for graphing lines and other shapes.

Q: How do I determine if a graph is a direct proportion?

A: To determine if a graph is a direct proportion, you can check if it is a straight line that passes through the origin (0, 0) and has a positive slope.

Q: What are some common mistakes to avoid when graphing a direct proportion?

A: Some common mistakes to avoid when graphing a direct proportion include:

Not using a ruler or other straightedge: This can lead to inaccurate graphing.
Not labeling the axes: This can make it difficult to understand the graph.
Not checking for a positive slope: This can indicate that the graph is not a direct proportion.

Q: Can I use a computer program to graph a direct proportion?

A: Yes, you can use a computer program to graph a direct proportion. Many computer programs, including graphing software and programming languages, have built-in functions for graphing lines and other shapes.

Q: How do I check if a graph is a direct proportion using a computer program?

A: To check if a graph is a direct proportion using a computer program, you can use the following steps:

Enter the equation: Enter the equation of the graph into the computer program.
Graph the equation: Use the computer program to graph the equation.
Check the slope: Check the slope of the graph to ensure that it is positive.
Check the intercepts: Check the intercepts of the graph to ensure that they are correct.

Q: What are some additional resources for learning about direct proportion and coordinate plane?

A: Some additional resources for learning about direct proportion and coordinate plane include:

Textbooks: Many textbooks cover direct proportion and coordinate plane in detail.
Online resources: Websites such as Khan Academy and Math Is Fun offer interactive lessons and exercises on direct proportion and coordinate plane.
Video tutorials: Video tutorials on YouTube and other platforms can provide additional help and support.