Using the translation that maps (3,-4) to its image (1,0), what is the image of any point (x,y)?

Given the translation that maps the point \((3, -4)\) to its image \((1, 0)\), we can determine the translation vector. Let's denote this vector by \((a, b)\).

The translation vector is found by subtracting the coordinates of the original point from the coordinates of the image point:
\[ (a, b) = (1 - 3, 0 - (-4)) \]
\[ (a, b) = (-2, 4) \]

This means the translation vector \((a, b) = (-2, 4)\) shifts any point \((x, y)\) by \(-2\) units in the \(x\)-direction and \(4\) units in the \(y\)-direction.

To find the image of any point \((x, y)\) under this translation, we apply the translation vector:
\[ (x', y') = (x + a, y + b) \]
\[ (x', y') = (x - 2, y + 4) \]

Therefore, the image of any point \((x, y)\) under the given translation is:
\[ (x - 2, y + 4) \]

If you have any questions or would like more details, feel free to ask.

Here are 8 related questions for further exploration:

1. What is the image of the point \((5, 7)\) under the same translation?
2. How would you describe the translation vector in terms of direction and magnitude?
3. Can you write the general form of the translation for any point \((x, y)\)?
4. What is the pre-image of the point \((2, 3)\) under this translation?
5. How would you find the translation vector if given another pair of original and image points?
6. How does this translation affect the shape and size of a geometric figure?
7. What is the effect of applying this translation twice to a point \((x, y)\)?
8. How can you represent this translation using matrix notation?

**Tip:** When dealing with translations, always pay attention to the direction and magnitude of the translation vector, as these determine how the points are shifted in the coordinate plane.

Explore how to determine the image of any point (x, y) under a translation that maps the point (3, -4) to its image (1, 0). Learn the translation vector, its application to any point (x, y), and related concepts in coordinate geometry. Suitable for middle school students.

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