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

A. (x+2,y−4)(𝑥+2,𝑦−4)

B. (x−2,y−4)(𝑥−2,𝑦−4)

C. (x−2,y+4)(𝑥−2,𝑦+4)

D. (x+2,y+4)


To determine the correct answer, we will revisit the steps to find the translation vector and how it is applied to any point \((x, y)\).

Given that the point \((3, -4)\) maps to the point \((1, 0)\), we first find the translation vector \((a, b)\) as follows:
\[ (a, b) = (1 - 3, 0 - (-4)) \]
\[ (a, b) = (-2, 4) \]

This translation vector indicates that any point \((x, y)\) is translated by \(-2\) units in the \(x\)-direction and \(4\) units in the \(y\)-direction. The translated point \((x', y')\) is given by:
\[ (x', y') = (x + a, y + b) \]
\[ (x', y') = (x - 2, y + 4) \]

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

Comparing this result with the given options:

A. \((x + 2, y - 4)\)

B. \((x - 2, y - 4)\)

C. \((x - 2, y + 4)\)

D. \((x + 2, y + 4)\)

The correct answer is:
C. \((x - 2, y + 4)\)

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

Here are 8 related questions for further exploration:

1. What would be the new coordinates of the point \((6, -2)\) under the same translation?
2. How do you reverse the translation to find the original point from its image?
3. How would you graphically represent the translation of a point in the coordinate plane?
4. What is the effect of this translation on the equation of a line?
5. If another translation maps \((3, -4)\) to \((5, -6)\), what would be the translation vector?
6. How would you apply this translation to a triangle with vertices \((0, 0)\), \((1, 0)\), and \((0, 1)\)?
7. How do you combine multiple translations into a single translation vector?
8. How would the translation affect the area and perimeter of a geometric shape?

**Tip:** When solving translation problems, carefully calculate the translation vector and apply it systematically to each coordinate to ensure accuracy.

Explore how to find the image of any point (x, y) under a translation that maps (3, -4) to (1, 0). Learn step-by-step with examples and compare options to choose the correct answer. Suitable for high school students.

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