Translation by 3 units to the right

Oct 13, 2025 by leolavanderia.com.br 33 views

Introduction

In mathematics, a coordinate plane is a fundamental concept used to represent points, lines, and shapes in a two-dimensional space. The coordinate plane is made up of two axes: the x-axis and the y-axis. Points on the coordinate plane are represented by ordered pairs (x, y), where x is the x-coordinate and y is the y-coordinate. In this article, we will explore the concept of translations on a coordinate plane and how to find a rule to describe the translation.

What is a Translation?

A translation is a transformation that moves a point or a shape from one location to another without changing its size or shape. In other words, a translation is a movement of a point or a shape in a specific direction and distance. Translations can be represented by a vector, which is a mathematical object that has both magnitude (length) and direction.

Step 1: Substitute the Original Coordinates and the Translated Coordinates

To find a rule to describe the translation, we need to substitute the original coordinates and the translated coordinates into the formula (x, y) → (x + h, y + k), where (h, k) is the translation vector. In this case, the original coordinates are (-6, 1) and the translated coordinates are (-3, -2).

Let's substitute the coordinates into the formula:

(-6, 1) → (-3, -2)

We can see that the x-coordinate has changed from -6 to -3, and the y-coordinate has changed from 1 to -2. This means that the translation vector is (h, k) = (-3 - (-6), -2 - 1) = (3, -3).

Step 2: Write the Translation Rule

Now that we have found the translation vector, we can write the translation rule in the form (x, y) → (x + h, y + k). In this case, the translation rule is:

(x, y) → (x + 3, y - 3)

This rule describes the translation that moves a point from (x, y) to (x + 3, y - 3).

Step 3: Test the Translation Rule

To test the translation rule, we can substitute a point into the rule and see if it produces the correct translated coordinates. Let's test the rule with the point (0, 0).

(0, 0) → (0 + 3, 0 - 3) = (3, -3)

We can see that the translated coordinates are indeed (3, -3), which is the correct result.

Conclusion

In this article, we have explored the concept of translations on a coordinate plane and how to find a rule to describe the translation. We have seen that a translation is a transformation that moves a point or a shape from one location to another without changing its size or shape. We have also seen how to substitute the original coordinates and the translated coordinates into the formula (x, y) → (x + h, y + k) to find the translation vector, and how to write the translation rule in the form (x, y) → (x + h, y + k). Finally, we have tested the translation rule with a point to ensure that it produces the correct translated coordinates.

Examples of Translations

Here are some examples of translations on a coordinate plane:

Translation by 3 units to the right: (x, y) → (x + 3, y)
Translation by 2 units down: (x, y) → (x, y - 2)
Translation by 4 units to the left: (x, y) → (x - 4, y)
Translation by 1 unit up: (x, y) → (x, y + 1)

Real-World Applications of Translations

Translations have many real-world applications in fields such as:

Computer Graphics: Translations are used to move objects on a screen or to create animations.
Architecture: Translations are used to design buildings and to create blueprints.
Engineering: Translations are used to design machines and to create prototypes.
Art: Translations are used to create art pieces and to design sculptures.

Conclusion

Q: What is a translation on a coordinate plane?

A: A translation on a coordinate plane is a transformation that moves a point or a shape from one location to another without changing its size or shape. In other words, a translation is a movement of a point or a shape in a specific direction and distance.

Q: How do I find the translation vector?

A: To find the translation vector, you need to substitute the original coordinates and the translated coordinates into the formula (x, y) → (x + h, y + k), where (h, k) is the translation vector.

Q: What is the formula for a translation?

A: The formula for a translation is (x, y) → (x + h, y + k), where (h, k) is the translation vector.

Q: How do I write the translation rule?

A: To write the translation rule, you need to substitute the translation vector into the formula (x, y) → (x + h, y + k).

Q: Can I test the translation rule with a point?

A: Yes, you can test the translation rule with a point to ensure that it produces the correct translated coordinates.

Q: What are some examples of translations on a coordinate plane?

A: Some examples of translations on a coordinate plane include:

Translation by 3 units to the right: (x, y) → (x + 3, y)
Translation by 2 units down: (x, y) → (x, y - 2)
Translation by 4 units to the left: (x, y) → (x - 4, y)
Translation by 1 unit up: (x, y) → (x, y + 1)

Q: What are some real-world applications of translations?

A: Some real-world applications of translations include:

Computer Graphics: Translations are used to move objects on a screen or to create animations.
Architecture: Translations are used to design buildings and to create blueprints.
Engineering: Translations are used to design machines and to create prototypes.
Art: Translations are used to create art pieces and to design sculptures.

Q: Can I use translations to solve problems in mathematics?

A: Yes, you can use translations to solve problems in mathematics, such as finding the coordinates of a translated point or determining the distance between two points.

Q: How do I determine the distance between two points after a translation?

A: To determine the distance between two points after a translation, you need to use the distance formula, which is √((x2 - x1)^2 + (y2 - y1)^2).

Q: Can I use translations to solve problems in science and engineering?

A: Yes, you can use translations to solve problems in science and engineering, such as designing machines or creating prototypes.

Q: How do I apply translations to real-world problems?

A: To apply translations to real-world problems, you need to identify the translation vector and use it to solve the problem.

Conclusion

In conclusion, translations are an important concept in mathematics that have many real-world applications. We have seen how to find the translation vector, write the translation rule, and test the translation rule with a point. We have also seen examples of translations and real-world applications of translations.