Jean Transformed A Point Using The Rule \[$(x, Y) \rightarrow (x-6, Y+8)\$\]. The Image Point Is \[$(-4, 1)\$\].Which Point Is The Original?A. \[$(-10, 9)\$\] B. \[$ (2, -7) \$\] C. \[$(-2, 7)\$\] D. \[$
Introduction
In mathematics, transformations are essential concepts that help us understand how to manipulate and change the position of points, shapes, and objects. One of the fundamental types of transformations is the point transformation, where a point is moved from one location to another based on a specific rule. In this article, we will explore how to transform points using a given rule and apply this knowledge to find the original point.
Understanding the Transformation Rule
The transformation rule given in the problem is {(x, y) \rightarrow (x-6, y+8)$}$. This rule states that for any point {(x, y)$}$, we need to subtract 6 from the x-coordinate and add 8 to the y-coordinate to get the new point. Let's break down this rule to understand it better.
- The x-coordinate is decreased by 6, which means the new x-coordinate is 6 units to the left of the original x-coordinate.
- The y-coordinate is increased by 8, which means the new y-coordinate is 8 units above the original y-coordinate.
Applying the Transformation Rule
Now that we understand the transformation rule, let's apply it to the given image point {(-4, 1)$}$. We need to substitute the values of x and y into the rule and calculate the new point.
- New x-coordinate:
- New y-coordinate:
Therefore, the new point after applying the transformation rule is {(-10, 9)$}$.
Finding the Original Point
The problem asks us to find the original point that was transformed to {(-10, 9)$}$ using the given rule. To do this, we need to work backwards and apply the inverse transformation rule.
-
Let's assume the original point is {(x, y)$}$.
-
We know that the new point is {(-10, 9)$}$, which means the new x-coordinate is -10 and the new y-coordinate is 9.
-
Using the transformation rule, we can set up the following equations:
Solving these equations will give us the original x and y coordinates.
-
Solving the first equation for x, we get:
-
Solving the second equation for y, we get:
Therefore, the original point is {(-4, 1)$}$.
Conclusion
In this article, we learned how to transform points using a given rule and applied this knowledge to find the original point. We understood the transformation rule, applied it to the given image point, and worked backwards to find the original point. This problem demonstrates the importance of understanding transformation rules and how to apply them to solve problems in mathematics.
Answer
Introduction
In our previous article, we explored how to transform points using a given rule and applied this knowledge to find the original point. In this article, we will answer some frequently asked questions related to transforming points and provide additional examples to help solidify your understanding.
Q: What is the purpose of transforming points?
A: Transforming points is an essential concept in mathematics that helps us understand how to manipulate and change the position of points, shapes, and objects. It is used in various fields such as geometry, algebra, and computer graphics.
Q: What is the difference between a transformation rule and a transformation matrix?
A: A transformation rule is a mathematical formula that describes how to transform a point from one location to another. A transformation matrix, on the other hand, is a matrix that represents a transformation and can be used to perform the transformation on a point or a shape.
Q: How do I determine the type of transformation (translation, rotation, scaling, etc.)?
A: To determine the type of transformation, you need to analyze the transformation rule or matrix. For example, if the rule involves adding or subtracting a constant value from the x or y coordinate, it is a translation. If the rule involves multiplying the x or y coordinate by a constant value, it is a scaling. If the rule involves changing the sign of the x or y coordinate, it is a reflection.
Q: Can I apply multiple transformations to a point or shape?
A: Yes, you can apply multiple transformations to a point or shape. The order in which you apply the transformations matters, and you need to follow the correct order to get the desired result.
Q: How do I find the inverse of a transformation rule or matrix?
A: To find the inverse of a transformation rule or matrix, you need to follow these steps:
- For a transformation rule, you need to swap the x and y coordinates and change the signs of the constants.
- For a transformation matrix, you need to take the inverse of the matrix.
Q: What is the difference between a rigid transformation and a non-rigid transformation?
A: A rigid transformation is a transformation that preserves the shape and size of the object, while a non-rigid transformation changes the shape and size of the object.
Q: Can I use transforming points to solve real-world problems?
A: Yes, transforming points is a powerful tool that can be used to solve real-world problems in various fields such as computer graphics, game development, and engineering.
Examples
Here are some examples of transforming points:
- Translation: If you want to move a point from (x, y) to (x + 3, y + 4), you can use the transformation rule (x, y) → (x + 3, y + 4).
- Scaling: If you want to scale a point by a factor of 2, you can use the transformation rule (x, y) → (2x, 2y).
- Rotation: If you want to rotate a point by 90 degrees clockwise, you can use the transformation rule (x, y) → (y, -x).
Conclusion
In this article, we answered some frequently asked questions related to transforming points and provided additional examples to help solidify your understanding. Transforming points is a powerful tool that can be used to solve real-world problems in various fields. By understanding the transformation rules and matrices, you can apply them to solve problems and create amazing visual effects.
Additional Resources
If you want to learn more about transforming points, here are some additional resources:
- Books: "Geometry: A Comprehensive Introduction" by Dan Pedoe, "Linear Algebra and Its Applications" by Gilbert Strang
- Online Courses: "Geometry" by 3Blue1Brown, "Linear Algebra" by MIT OpenCourseWare
- Tutorials: "Transforming Points" by Khan Academy, "Transformation Matrices" by Math Is Fun