Determining Translations On The Coordinate PlaneWhat Is The \[$ Y \$\]-coordinate Of Point \[$ D \$\] After A Translation Of \[$(x, Y) \rightarrow (x+6, Y-4)\$\]?\[$ D^{\prime}(3.5, \square) \$\]
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Understanding Coordinate Translations
In mathematics, a translation is a fundamental concept that involves shifting a point or an object from one location to another on a coordinate plane. This process involves changing the coordinates of the point, and it is essential to understand how to determine the new coordinates after a translation. In this article, we will explore the concept of coordinate translations and learn how to determine the new coordinates of a point after a translation.
What is a Coordinate Translation?
A coordinate translation is a transformation that involves shifting a point or an object from one location to another on a coordinate plane. This process involves changing the coordinates of the point, and it is essential to understand how to determine the new coordinates after a translation. A translation can be represented by the equation (x, y) → (x + a, y + b), where (x, y) is the original point, and (x + a, y + b) is the new point after the translation.
Determining the New Coordinates
To determine the new coordinates of a point after a translation, we need to understand how the translation affects the x and y coordinates. The x-coordinate is shifted by the value of 'a', and the y-coordinate is shifted by the value of 'b'. Therefore, if we have a point (x, y) and we want to translate it by (a, b), the new coordinates will be (x + a, y + b).
Example: Translating a Point on the Coordinate Plane
Let's consider an example to understand how to determine the new coordinates after a translation. Suppose we have a point D(3.5, 2) and we want to translate it by (6, -4). To determine the new coordinates of point D, we need to add 6 to the x-coordinate and subtract 4 from the y-coordinate.
Calculating the New Coordinates
To calculate the new coordinates of point D, we need to add 6 to the x-coordinate and subtract 4 from the y-coordinate.
x-coordinate: 3.5 + 6 = 9.5 y-coordinate: 2 - 4 = -2
Therefore, the new coordinates of point D are (9.5, -2).
Conclusion
In conclusion, determining the new coordinates after a translation is a crucial concept in mathematics. By understanding how the translation affects the x and y coordinates, we can determine the new coordinates of a point after a translation. In this article, we learned how to determine the new coordinates of a point after a translation and applied this concept to an example problem.
Frequently Asked Questions
Q: What is a coordinate translation?
A: A coordinate translation is a transformation that involves shifting a point or an object from one location to another on a coordinate plane.
Q: How do I determine the new coordinates after a translation?
A: To determine the new coordinates after a translation, you need to add the value of 'a' to the x-coordinate and add the value of 'b' to the y-coordinate.
Q: What is the formula for a coordinate translation?
A: The formula for a coordinate translation is (x, y) → (x + a, y + b), where (x, y) is the original point, and (x + a, y + b) is the new point after the translation.
Additional Resources
- Khan Academy: Coordinate Geometry
- Math Open Reference: Coordinate Geometry
- Wolfram MathWorld: Coordinate Geometry
Final Thoughts
Determining the new coordinates after a translation is a fundamental concept in mathematics. By understanding how the translation affects the x and y coordinates, we can determine the new coordinates of a point after a translation. In this article, we learned how to determine the new coordinates of a point after a translation and applied this concept to an example problem. We hope this article has provided you with a better understanding of coordinate translations and how to determine the new coordinates after a translation.
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Understanding Coordinate Translations: Q&A
In our previous article, we explored the concept of coordinate translations and learned how to determine the new coordinates of a point after a translation. In this article, we will answer some frequently asked questions about coordinate translations and provide additional resources for further learning.
Q&A: Coordinate Translations
Q: What is the difference between a translation and a rotation?
A: A translation is a transformation that involves shifting a point or an object from one location to another on a coordinate plane, while a rotation is a transformation that involves rotating a point or an object around a fixed point.
Q: How do I determine the new coordinates after a translation?
A: To determine the new coordinates after a translation, you need to add the value of 'a' to the x-coordinate and add the value of 'b' to the y-coordinate.
Q: What is the formula for a coordinate translation?
A: The formula for a coordinate translation is (x, y) → (x + a, y + b), where (x, y) is the original point, and (x + a, y + b) is the new point after the translation.
Q: Can I translate a point by a negative value?
A: Yes, you can translate a point by a negative value. For example, if you want to translate a point by (-3, 2), you would subtract 3 from the x-coordinate and add 2 to the y-coordinate.
Q: How do I determine the new coordinates after a translation if the translation is not a whole number?
A: To determine the new coordinates after a translation if the translation is not a whole number, you can simply add the decimal value to the x-coordinate and add the decimal value to the y-coordinate.
Q: Can I translate a point by a fraction?
A: Yes, you can translate a point by a fraction. For example, if you want to translate a point by (1/2, 3/4), you would add 1/2 to the x-coordinate and add 3/4 to the y-coordinate.
Example Problems: Coordinate Translations
Problem 1: Translating a Point by a Whole Number
Translate the point (2, 3) by (4, 2).
x-coordinate: 2 + 4 = 6 y-coordinate: 3 + 2 = 5
Therefore, the new coordinates of the point are (6, 5).
Problem 2: Translating a Point by a Decimal Value
Translate the point (2, 3) by (2.5, 1.2).
x-coordinate: 2 + 2.5 = 4.5 y-coordinate: 3 + 1.2 = 4.2
Therefore, the new coordinates of the point are (4.5, 4.2).
Problem 3: Translating a Point by a Fraction
Translate the point (2, 3) by (1/2, 3/4).
x-coordinate: 2 + 1/2 = 2.5 y-coordinate: 3 + 3/4 = 3.75
Therefore, the new coordinates of the point are (2.5, 3.75).
Conclusion
In conclusion, determining the new coordinates after a translation is a crucial concept in mathematics. By understanding how the translation affects the x and y coordinates, we can determine the new coordinates of a point after a translation. In this article, we answered some frequently asked questions about coordinate translations and provided additional resources for further learning.
Frequently Asked Questions
Q: What is the difference between a translation and a rotation?
A: A translation is a transformation that involves shifting a point or an object from one location to another on a coordinate plane, while a rotation is a transformation that involves rotating a point or an object around a fixed point.
Q: How do I determine the new coordinates after a translation?
A: To determine the new coordinates after a translation, you need to add the value of 'a' to the x-coordinate and add the value of 'b' to the y-coordinate.
Q: What is the formula for a coordinate translation?
A: The formula for a coordinate translation is (x, y) → (x + a, y + b), where (x, y) is the original point, and (x + a, y + b) is the new point after the translation.
Additional Resources
- Khan Academy: Coordinate Geometry
- Math Open Reference: Coordinate Geometry
- Wolfram MathWorld: Coordinate Geometry
Final Thoughts
Determining the new coordinates after a translation is a fundamental concept in mathematics. By understanding how the translation affects the x and y coordinates, we can determine the new coordinates of a point after a translation. In this article, we answered some frequently asked questions about coordinate translations and provided additional resources for further learning. We hope this article has provided you with a better understanding of coordinate translations and how to determine the new coordinates after a translation.