A Rectangle On A Coordinate Plane Is Translated 5 Units Up And 3 Units To The Left. Which Rule Describes The Translation?A. \[$(x, Y) \rightarrow (x-3, Y+5)\$\]B. \[$(x, Y) \rightarrow (x+5, Y+3)\$\]C. \[$(x, Y) \rightarrow (x+3,

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Introduction

In mathematics, translations are an essential concept in geometry, particularly when working with coordinate planes. A translation is a transformation that moves a figure from one location to another without changing its size or shape. In this article, we will explore the concept of translations on a coordinate plane and determine which rule describes a specific translation.

What is a Translation?

A translation is a transformation that moves a figure from one location to another without changing its size or shape. It is a type of rigid motion that preserves the distance and angle between points. In a coordinate plane, a translation can be represented by a vector, which is an ordered pair of numbers that indicates the direction and magnitude of the movement.

Notation for Translations

When representing a translation on a coordinate plane, we use the notation (x,y)→(x′,y′)(x, y) \rightarrow (x', y'), where (x,y)(x, y) is the original point and (x′,y′)(x', y') is the translated point. The arrow →\rightarrow indicates the direction of the translation.

Rule A: (x,y)→(x−3,y+5)(x, y) \rightarrow (x-3, y+5)

Let's analyze Rule A: (x,y)→(x−3,y+5)(x, y) \rightarrow (x-3, y+5). This rule states that the x-coordinate of the original point is decreased by 3 units, and the y-coordinate is increased by 5 units. In other words, the point is moved 3 units to the left and 5 units up.

Rule B: (x,y)→(x+5,y+3)(x, y) \rightarrow (x+5, y+3)

Now, let's examine Rule B: (x,y)→(x+5,y+3)(x, y) \rightarrow (x+5, y+3). This rule states that the x-coordinate of the original point is increased by 5 units, and the y-coordinate is increased by 3 units. In other words, the point is moved 5 units to the right and 3 units up.

Rule C: (x,y)→(x+3,y−5)(x, y) \rightarrow (x+3, y-5)

Finally, let's consider Rule C: (x,y)→(x+3,y−5)(x, y) \rightarrow (x+3, y-5). This rule states that the x-coordinate of the original point is increased by 3 units, and the y-coordinate is decreased by 5 units. In other words, the point is moved 3 units to the right and 5 units down.

Which Rule Describes the Translation?

Now that we have analyzed each rule, let's determine which one describes the translation of moving a rectangle 5 units up and 3 units to the left. Based on our analysis, we can see that Rule A: (x,y)→(x−3,y+5)(x, y) \rightarrow (x-3, y+5) is the correct rule. This rule states that the x-coordinate of the original point is decreased by 3 units, and the y-coordinate is increased by 5 units, which corresponds to moving the rectangle 3 units to the left and 5 units up.

Conclusion

In conclusion, translations are an essential concept in geometry, particularly when working with coordinate planes. By understanding the notation for translations and analyzing each rule, we can determine which rule describes a specific translation. In this article, we have explored the concept of translations on a coordinate plane and determined that Rule A: (x,y)→(x−3,y+5)(x, y) \rightarrow (x-3, y+5) describes the translation of moving a rectangle 5 units up and 3 units to the left.

Example Problems

  1. A point is translated 2 units up and 4 units to the right. Which rule describes the translation?
  2. A rectangle is translated 3 units down and 2 units to the left. Which rule describes the translation?
  3. A point is translated 5 units up and 1 unit to the right. Which rule describes the translation?

Answer Key

  1. (x,y)→(x+4,y+2)(x, y) \rightarrow (x+4, y+2)
  2. (x,y)→(x−2,y−3)(x, y) \rightarrow (x-2, y-3)
  3. (x,y)→(x+1,y+5)(x, y) \rightarrow (x+1, y+5)

Practice Problems

  1. A point is translated 1 unit up and 3 units to the left. Write the rule that describes the translation.
  2. A rectangle is translated 2 units down and 4 units to the right. Write the rule that describes the translation.
  3. A point is translated 5 units up and 2 units to the left. Write the rule that describes the translation.

Answer Key

  1. (x,y)→(x−3,y+1)(x, y) \rightarrow (x-3, y+1)
  2. (x,y)→(x+4,y−2)(x, y) \rightarrow (x+4, y-2)
  3. (x,y)→(x−2,y+5)(x, y) \rightarrow (x-2, y+5)
    Coordinate Plane Translations: Q&A =====================================

Introduction

In our previous article, we explored the concept of translations on a coordinate plane and determined which rule describes a specific translation. In this article, we will answer some frequently asked questions about coordinate plane translations.

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

A: A translation is a transformation that moves a figure from one location to another without changing its size or shape. A rotation, on the other hand, is a transformation that turns a figure around a fixed point without changing its size or shape.

Q: How do I determine the rule for a translation?

A: To determine the rule for a translation, you need to identify the direction and magnitude of the movement. If the point is moved to the right, the x-coordinate increases. If the point is moved to the left, the x-coordinate decreases. If the point is moved up, the y-coordinate increases. If the point is moved down, the y-coordinate decreases.

Q: What is the notation for a translation?

A: The notation for a translation is (x,y)→(x′,y′)(x, y) \rightarrow (x', y'), where (x,y)(x, y) is the original point and (x′,y′)(x', y') is the translated point.

Q: Can a translation be represented by a vector?

A: Yes, a translation can be represented by a vector. A vector is an ordered pair of numbers that indicates the direction and magnitude of the movement.

Q: How do I write the rule for a translation?

A: To write the rule for a translation, you need to identify the direction and magnitude of the movement and write the corresponding change in the x and y coordinates.

Q: What is the difference between a translation and a reflection?

A: A translation is a transformation that moves a figure from one location to another without changing its size or shape. A reflection, on the other hand, is a transformation that flips a figure over a line without changing its size or shape.

Q: Can a translation be represented by a matrix?

A: Yes, a translation can be represented by a matrix. A matrix is a rectangular array of numbers that can be used to represent a transformation.

Q: How do I determine the matrix for a translation?

A: To determine the matrix for a translation, you need to identify the direction and magnitude of the movement and write the corresponding matrix.

Q: What is the relationship between a translation and a vector?

A: A translation can be represented by a vector, and a vector can be used to represent a translation.

Q: Can a translation be represented by a function?

A: Yes, a translation can be represented by a function. A function is a relation between a set of inputs and a set of possible outputs.

Q: How do I write the function for a translation?

A: To write the function for a translation, you need to identify the direction and magnitude of the movement and write the corresponding function.

Conclusion

In conclusion, coordinate plane translations are an essential concept in geometry, and understanding the notation, rules, and relationships between translations and other transformations is crucial for success in mathematics. By answering these frequently asked questions, we hope to have provided a better understanding of coordinate plane translations.

Practice Problems

  1. A point is translated 2 units up and 4 units to the right. Write the rule that describes the translation.
  2. A rectangle is translated 3 units down and 2 units to the left. Write the rule that describes the translation.
  3. A point is translated 5 units up and 1 unit to the right. Write the rule that describes the translation.

Answer Key

  1. (x,y)→(x+4,y+2)(x, y) \rightarrow (x+4, y+2)
  2. (x,y)→(x−2,y−3)(x, y) \rightarrow (x-2, y-3)
  3. (x,y)→(x+1,y+5)(x, y) \rightarrow (x+1, y+5)

Additional Resources