Which Rule Describes A Composition That Maps Pre-image PQRS?A. R 0 , 270 ∘ ∘ T − 2 , 0 ( X , Y R_{0,270^{\circ}} \circ T_{-2,0}(x, Y R 0 , 27 0 ∘ ​ ∘ T − 2 , 0 ​ ( X , Y ] B. T − 2 , 0 ∘ R 0 , 270 ∘ ( X , Y T_{-2,0} \circ R_{0,270^{\circ}}(x, Y T − 2 , 0 ​ ∘ R 0 , 27 0 ∘ ​ ( X , Y ] C. R 0 , 270 ∘ ∘ R Y -axis ( X , Y R_{0,270^{\circ}} \circ R_{y\text{-axis}}(x, Y R 0 , 27 0 ∘ ​ ∘ R Y -axis ​ ( X , Y ] D.

Introduction

Transformations are a fundamental concept in mathematics, particularly in geometry and algebra. They involve changing the position, size, or orientation of a figure or shape. In this article, we will explore the concept of transformations and how they are represented using mathematical notation. We will also examine the rules that describe specific transformations and how they can be combined to create more complex transformations.

What are Transformations?

Transformations are functions that take a figure or shape as input and produce a new figure or shape as output. They can be thought of as a way of "moving" or "changing" a figure in some way. There are several types of transformations, including:

• Translation: A translation is a transformation that moves a figure a certain distance in a specific direction.
• Rotation: A rotation is a transformation that turns a figure around a fixed point.
• Reflection: A reflection is a transformation that flips a figure over a line or plane.
• Dilation: A dilation is a transformation that changes the size of a figure.

Representing Transformations Mathematically

Transformations can be represented mathematically using a variety of notations. The most common notation is the function notation, where the transformation is represented as a function of the input coordinates. For example, the translation transformation $T_{-2,0}(x, y)$ can be represented as:

$$T_{-2,0}(x, y) = (x - 2, y)$$

Similarly, the rotation transformation $R_{0,270^{\circ}}(x, y)$ can be represented as:

$$R_{0,270^{\circ}}(x, y) = (-y, x)$$

Combining Transformations

Transformations can be combined to create more complex transformations. This is known as the composition of transformations. The composition of two transformations $T_1$ and $T_2$ is denoted as $T_2 \circ T_1$. For example, the composition of the translation transformation $T_{-2,0}(x, y)$ and the rotation transformation $R_{0,270^{\circ}}(x, y)$ can be represented as:

$$R_{0,270^{\circ}} \circ T_{-2,0}(x, y) = R_{0,270^{\circ}}(x - 2, y) = (-y, x - 2)$$

Which Rule Describes a Composition that Maps Pre-Image PQRS?

Now that we have a good understanding of transformations and how they can be represented mathematically, let's examine the options provided in the problem statement.

A. $R_{0,270^{\circ}} \circ T_{-2,0}(x, y)$

This option represents the composition of the rotation transformation $R_{0,270^{\circ}}(x, y)$ and the translation transformation $T_{-2,0}(x, y)$. This means that the pre-image PQRS will be rotated by $270^{\circ}$ and then translated by $-2$ units in the x-direction.

B. $T_{-2,0} \circ R_{0,270^{\circ}}(x, y)$

This option represents the composition of the translation transformation $T_{-2,0}(x, y)$ and the rotation transformation $R_{0,270^{\circ}}(x, y)$. This means that the pre-image PQRS will be rotated by $270^{\circ}$ and then translated by $-2$ units in the x-direction.

C. $R_{0,270^{\circ}} \circ r_{y\text{-axis}}(x, y)$

This option represents the composition of the rotation transformation $R_{0,270^{\circ}}(x, y)$ and the reflection transformation $r_{y\text{-axis}}(x, y)$. This means that the pre-image PQRS will be reflected over the y-axis and then rotated by $270^{\circ}$.

D. $r_{y\text{-axis}} \circ R_{0,270^{\circ}}(x, y)$

This option represents the composition of the reflection transformation $r_{y\text{-axis}}(x, y)$ and the rotation transformation $R_{0,270^{\circ}}(x, y)$. This means that the pre-image PQRS will be rotated by $270^{\circ}$ and then reflected over the y-axis.

Conclusion

In conclusion, the correct answer to the problem statement is option A, $R_{0,270^{\circ}} \circ T_{-2,0}(x, y)$. This option represents the composition of the rotation transformation $R_{0,270^{\circ}}(x, y)$ and the translation transformation $T_{-2,0}(x, y)$, which maps the pre-image PQRS to the desired image.

Final Answer

Introduction

Transformations are a fundamental concept in mathematics, particularly in geometry and algebra. They involve changing the position, size, or orientation of a figure or shape. In this article, we will explore the concept of transformations and answer some frequently asked questions about them.

Q: What is a transformation?

A: A transformation is a function that takes a figure or shape as input and produces a new figure or shape as output. It can be thought of as a way of "moving" or "changing" a figure in some way.

Q: What are the different types of transformations?

A: There are several types of transformations, including:

• Translation: A translation is a transformation that moves a figure a certain distance in a specific direction.
• Rotation: A rotation is a transformation that turns a figure around a fixed point.
• Reflection: A reflection is a transformation that flips a figure over a line or plane.
• Dilation: A dilation is a transformation that changes the size of a figure.

Q: How are transformations represented mathematically?

A: Transformations can be represented mathematically using a variety of notations. The most common notation is the function notation, where the transformation is represented as a function of the input coordinates.

Q: What is the composition of transformations?

A: The composition of transformations is the process of combining two or more transformations to create a new transformation. This is denoted as $T_2 \circ T_1$, where $T_1$ and $T_2$ are the individual transformations.

Q: How do I determine the order of transformations?

A: The order of transformations is important because it determines the final result of the composition. In general, the order of transformations is from left to right, meaning that the first transformation is applied first, followed by the second transformation, and so on.

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

A: A translation is a transformation that moves a figure a certain distance in a specific direction, while a rotation is a transformation that turns a figure around a fixed point.

Q: Can I combine multiple transformations to create a new transformation?

A: Yes, you can combine multiple transformations to create a new transformation. This is known as the composition of transformations.

Q: How do I apply a transformation to a figure?

A: To apply a transformation to a figure, you need to follow these steps:

1. Identify the transformation you want to apply.
2. Determine the order of the transformation.
3. Apply the transformation to the figure.

Q: What are some common applications of transformations?

A: Transformations have many applications in mathematics, science, and engineering. Some common applications include:

• Geometry: Transformations are used to study the properties of geometric shapes and figures.
• Algebra: Transformations are used to solve equations and inequalities.
• Computer Graphics: Transformations are used to create 3D models and animations.
• Engineering: Transformations are used to design and analyze mechanical systems.

Conclusion

In conclusion, transformations are a fundamental concept in mathematics that involve changing the position, size, or orientation of a figure or shape. By understanding the different types of transformations, how they are represented mathematically, and how to apply them to figures, you can solve a wide range of problems in mathematics, science, and engineering.

Final Answer

The final answer is that transformations are a powerful tool in mathematics that can be used to solve a wide range of problems. By understanding the different types of transformations and how to apply them, you can unlock new insights and solutions in mathematics, science, and engineering.