The Image Of A Point Is Given By The Rule $r_{y=-x}(x, Y) \rightarrow (-4, 9$\]. What Are The Coordinates Of Its Pre-image?A. $(-9, 4$\] B. $(-4, -9$\] C. $(4, 9$\] D. $(9, -4$\]

Introduction

In mathematics, particularly in geometry and coordinate geometry, the concept of reflection is a fundamental idea that deals with the transformation of points from one position to another. When a point is reflected over a line, its image is formed, and the original point is known as the pre-image. In this article, we will explore the concept of the image of a point and its pre-image, focusing on the given rule $r_{y=-x}(x, y) \rightarrow (-4, 9)$ and determining the coordinates of its pre-image.

Understanding the Reflection Rule

The given rule $r_{y=-x}(x, y) \rightarrow (-4, 9)$ represents a reflection over the line $y = -x$. This line is a diagonal line that passes through the origin and has a slope of -1. When a point is reflected over this line, its $x$-coordinate becomes the negative of its $y$-coordinate, and its $y$-coordinate becomes the negative of its $x$-coordinate.

The Image of a Point

To find the image of a point $(x, y)$ under the reflection rule $r_{y=-x}$, we need to apply the transformation. The new coordinates of the image point are given by:

$$(-y, -x)$$

Using this transformation, we can find the image of any point $(x, y)$.

The Pre-Image of a Point

The pre-image of a point is the original point that is reflected to form the image. To find the pre-image of a point $(x', y')$, we need to apply the inverse transformation of the reflection rule. The original coordinates of the pre-image point are given by:

$$(y', -x')$$

Using this transformation, we can find the pre-image of any point $(x', y')$.

Finding the Pre-Image of the Given Image

Now, let's apply the inverse transformation to find the pre-image of the given image point $(-4, 9)$. We need to find the original coordinates $(x, y)$ that are reflected to form the image point $(-4, 9)$.

Using the inverse transformation, we get:

$$(y', -x') = (9, -(-4))$$

$$(y', -x') = (9, 4)$$

Therefore, the pre-image of the given image point $(-4, 9)$ is the point $(4, -9)$.

Conclusion

In this article, we explored the concept of the image of a point and its pre-image, focusing on the given rule $r_{y=-x}(x, y) \rightarrow (-4, 9)$. We applied the reflection rule and its inverse transformation to find the pre-image of the given image point. The pre-image of the given image point $(-4, 9)$ is the point $(4, -9)$. This concept is essential in mathematics, particularly in geometry and coordinate geometry, and has numerous applications in various fields.

Frequently Asked Questions

• What is the reflection rule $r_{y=-x}$?
  • The reflection rule $r_{y=-x}$ represents a reflection over the line $y = -x$.
• How do you find the image of a point under the reflection rule $r_{y=-x}$?
  • To find the image of a point $(x, y)$ under the reflection rule $r_{y=-x}$, we need to apply the transformation: $(-y, -x)$.
• How do you find the pre-image of a point under the reflection rule $r_{y=-x}$?
  • To find the pre-image of a point $(x', y')$ under the reflection rule $r_{y=-x}$, we need to apply the inverse transformation: $(y', -x')$.

Final Answer

The pre-image of the given image point $(-4, 9)$ is the point $(4, -9)$.

Introduction

In our previous article, we explored the concept of the image of a point and its pre-image, focusing on the given rule $r_{y=-x}(x, y) \rightarrow (-4, 9)$. We applied the reflection rule and its inverse transformation to find the pre-image of the given image point. In this article, we will provide a Q&A section to further clarify the concept and answer any questions that readers may have.

Q&A

Q1: What is the reflection rule $r_{y=-x}$?

A1: The reflection rule $r_{y=-x}$ represents a reflection over the line $y = -x$. This line is a diagonal line that passes through the origin and has a slope of -1.

Q2: How do you find the image of a point under the reflection rule $r_{y=-x}$?

A2: To find the image of a point $(x, y)$ under the reflection rule $r_{y=-x}$, we need to apply the transformation: $(-y, -x)$.

Q3: How do you find the pre-image of a point under the reflection rule $r_{y=-x}$?

A3: To find the pre-image of a point $(x', y')$ under the reflection rule $r_{y=-x}$, we need to apply the inverse transformation: $(y', -x')$.

Q4: What is the difference between the image and the pre-image of a point?

A4: The image of a point is the point that is formed after applying the reflection rule, while the pre-image of a point is the original point that is reflected to form the image.

Q5: Can you provide an example of finding the pre-image of a point?

A5: Let's consider the point $(-3, 4)$. To find its pre-image, we need to apply the inverse transformation: $(y', -x') = (4, -(-3)) = (4, 3)$. Therefore, the pre-image of the point $(-3, 4)$ is the point $(3, -4)$.

Q6: How do you determine the coordinates of the pre-image of a point?

A6: To determine the coordinates of the pre-image of a point, we need to apply the inverse transformation: $(y', -x')$. This will give us the original coordinates of the pre-image point.

Q7: Can you provide an example of finding the image of a point?

A7: Let's consider the point $(2, -5)$. To find its image, we need to apply the transformation: $(-y, -x) = (-(-5), -2) = (5, -2)$. Therefore, the image of the point $(2, -5)$ is the point $(5, -2)$.

Q8: How do you determine the coordinates of the image of a point?

A8: To determine the coordinates of the image of a point, we need to apply the transformation: $(-y, -x)$. This will give us the new coordinates of the image point.

Conclusion

In this Q&A article, we provided answers to common questions about the image of a point and its pre-image. We clarified the concept of the reflection rule $r_{y=-x}$ and its inverse transformation. We also provided examples of finding the pre-image and image of a point. We hope that this article has been helpful in further clarifying the concept and answering any questions that readers may have.

Frequently Asked Questions

• What is the reflection rule $r_{y=-x}$?
  • The reflection rule $r_{y=-x}$ represents a reflection over the line $y = -x$.
• How do you find the image of a point under the reflection rule $r_{y=-x}$?
  • To find the image of a point $(x, y)$ under the reflection rule $r_{y=-x}$, we need to apply the transformation: $(-y, -x)$.
• How do you find the pre-image of a point under the reflection rule $r_{y=-x}$?
  • To find the pre-image of a point $(x', y')$ under the reflection rule $r_{y=-x}$, we need to apply the inverse transformation: $(y', -x')$.

Final Answer

The pre-image of the given image point $(-4, 9)$ is the point $(4, -9)$.