A Figure Is Translated Using The Mapping { (x, Y) \rightarrow (x+a, Y+b)$}$.If The Value Of { A$}$ Is Negative And The Value Of { B$}$ Is Negative, Which Best Describes The Translation?A. The Figure Moves Left And Up.B.

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Introduction

In mathematics, a translation is a fundamental concept in geometry that involves moving a figure from one position to another without changing its size or orientation. The translation is defined by a mapping function, which in this case is given as {(x, y) \rightarrow (x+a, y+b)$}$. This function takes a point {(x, y)$}$ in the original position and maps it to a new point {(x+a, y+b)$}$ in the translated position. The values of {a$}$ and {b$}$ determine the direction and magnitude of the translation.

Understanding the Translation Mapping

The translation mapping {(x, y) \rightarrow (x+a, y+b)$}$ can be broken down into two components: the horizontal translation and the vertical translation. The horizontal translation is given by the value of {a$}$, which represents the change in the x-coordinate. If {a$}$ is positive, the figure moves to the right, while if {a$}$ is negative, the figure moves to the left. Similarly, the vertical translation is given by the value of {b$}$, which represents the change in the y-coordinate. If {b$}$ is positive, the figure moves down, while if {b$}$ is negative, the figure moves up.

Analyzing the Given Conditions

Given that the value of {a$}$ is negative and the value of {b$}$ is negative, we can analyze the translation as follows:

  • Since {a$}$ is negative, the figure moves to the left.
  • Since {b$}$ is negative, the figure moves up.

Conclusion

Based on the analysis, the best description of the translation is that the figure moves left and up.

Example

To illustrate this concept, consider a square with vertices at {(0, 0)\$, \[(2, 0)\$, \[(2, 2)\$, and \[(0, 2)\$. If we apply the translation mapping \[(x, y) \rightarrow (x-2, y-2)$}$ to this square, the new vertices will be at ${(-2, -2)$, [0, -2)$, [0, 0)$, and [(-2, 0)$. As expected, the square has moved left and up.

Discussion

The translation mapping [(x, y) \rightarrow (x+a, y+b)\$} is a fundamental concept in geometry that can be used to describe the movement of figures in a coordinate plane. By analyzing the values of {a$}$ and {b$}$, we can determine the direction and magnitude of the translation. In this case, we have shown that if {a$}$ is negative and {b$}$ is negative, the figure moves left and up.

Applications

Translations have numerous applications in various fields, including:

  • Computer graphics: Translations are used to move objects in a 2D or 3D space.
  • Engineering: Translations are used to describe the movement of mechanical systems.
  • Physics: Translations are used to describe the motion of objects in a coordinate plane.

Exercises

  1. If the value of {a$}$ is positive and the value of {b$}$ is negative, which best describes the translation?
  2. If the value of {a$}$ is negative and the value of {b$}$ is positive, which best describes the translation?
  3. If the value of {a$}$ is zero and the value of {b$}$ is negative, which best describes the translation?

Solutions

  1. The figure moves right and up.
  2. The figure moves left and down.
  3. The figure moves up.

Conclusion

In conclusion, the translation mapping {(x, y) \rightarrow (x+a, y+b)$}$ is a fundamental concept in geometry that can be used to describe the movement of figures in a coordinate plane. By analyzing the values of {a$}$ and {b$}$, we can determine the direction and magnitude of the translation. In this case, we have shown that if {a$}$ is negative and {b$}$ is negative, the figure moves left and up.

Introduction

In our previous article, we discussed the translation mapping {(x, y) \rightarrow (x+a, y+b)$}$ and analyzed the case where the value of {a$}$ is negative and the value of {b$}$ is negative. We concluded that the figure moves left and up. In this article, we will provide a Q&A section to further clarify the concept of translation and answer some common questions.

Q&A

Q1: What is the difference between a translation and a rotation?

A1: A translation is a movement of a figure from one position to another without changing its size or orientation. A rotation, on the other hand, is a movement of a figure around a fixed point, resulting in a change in its orientation.

Q2: How do I determine the direction of the translation?

A2: To determine the direction of the translation, you need to analyze the values of {a$}$ and {b$}$. If {a$}$ is positive, the figure moves to the right. If {a$}$ is negative, the figure moves to the left. If {b$}$ is positive, the figure moves down. If {b$}$ is negative, the figure moves up.

Q3: Can a translation be represented by a single value?

A3: No, a translation cannot be represented by a single value. A translation is represented by two values, {a$}$ and {b$}$, which determine the direction and magnitude of the translation.

Q4: How do I apply a translation to a figure?

A4: To apply a translation to a figure, you need to add the values of {a$}$ and {b$}$ to the coordinates of each point on the figure. For example, if the original coordinates are ${(x, y)$, the new coordinates will be [(x+a, y+b)$.

Q5: Can a translation be combined with other transformations?

A5: Yes, a translation can be combined with other transformations, such as rotations and reflections. However, the order of the transformations is important, as it can affect the final result.

Q6: How do I determine the magnitude of the translation?

A6: The magnitude of the translation is determined by the values of [a\$} and {b$}$. The magnitude is the distance between the original and final positions of the figure.

Q7: Can a translation be represented by a matrix?

A7: Yes, a translation can be represented by a matrix. The matrix is given by:

[10a01b001]{ \begin{bmatrix} 1 & 0 & a \\ 0 & 1 & b \\ 0 & 0 & 1 \end{bmatrix} }

Q8: How do I apply a translation to a 3D figure?

A8: To apply a translation to a 3D figure, you need to add the values of {a$}$ and {b$}$ to the coordinates of each point on the figure, taking into account the z-coordinate.

Conclusion

In conclusion, the translation mapping {(x, y) \rightarrow (x+a, y+b)$}$ is a fundamental concept in geometry that can be used to describe the movement of figures in a coordinate plane. By analyzing the values of {a$}$ and {b$}$, we can determine the direction and magnitude of the translation. We hope that this Q&A section has provided a better understanding of the concept of translation and its applications.

Exercises

  1. If the value of {a$}$ is positive and the value of {b$}$ is negative, which best describes the translation?
  2. If the value of {a$}$ is negative and the value of {b$}$ is positive, which best describes the translation?
  3. If the value of {a$}$ is zero and the value of {b$}$ is negative, which best describes the translation?

Solutions

  1. The figure moves right and up.
  2. The figure moves left and down.
  3. The figure moves up.

Discussion

Translations have numerous applications in various fields, including computer graphics, engineering, and physics. By understanding the concept of translation, we can better analyze and solve problems in these fields.

Applications

Translations have numerous applications in various fields, including:

  • Computer graphics: Translations are used to move objects in a 2D or 3D space.
  • Engineering: Translations are used to describe the movement of mechanical systems.
  • Physics: Translations are used to describe the motion of objects in a coordinate plane.

Further Reading

For further reading on the concept of translation, we recommend the following resources:

  • "Geometry: A Comprehensive Introduction" by Dan Pedoe
  • "Linear Algebra and Its Applications" by Gilbert Strang
  • "Computer Graphics: Principles and Practice" by James D. Foley and others

We hope that this article has provided a better understanding of the concept of translation and its applications.