On a coordinate plane, 5 squares are shown. Square L M N P has points (negative 3, 1), (negative 1, 1), (negative 1, negative 1), (negative 3, negative 1). Square 1 has points (negative 6, 4), (negative 6, 6), (negative 4, 6), (negative 4, 6). Square 2 has points (negative 6, negative 4), (negative 6, negative 6), (negative 4, negative 4), (negative 4, negative 6). Square 3 has points (2, 2), (2, 4), (4, 4), (4, 2). Square 4 has points (2, negative 2), (4, negative 2), (4, negative 4), (2, negative 4).

Question

Assume each figure shown has the same orientation. Which figure is the image of square LMNP after a translation of
(x, y) → (x + 5, y – 3)?

Answer 1

To find the image of square LMNP after the translation defined by the function \( (x, y) \to (x + 5, y - 3) \), we will apply this translation to each of the vertices of square LMNP.

The vertices of square LMNP are:

• \( L(-3, 1) \)
• \( M(-1, 1) \)
• \( N(-1, -1) \)
• \( P(-3, -1) \)

Now, applying the translation to each vertex:

1. For point \( L(-3, 1) \): \[ L' = (-3 + 5, 1 - 3) = (2, -2) \]

2. For point \( M(-1, 1) \): \[ M' = (-1 + 5, 1 - 3) = (4, -2) \]

3. For point \( N(-1, -1) \): \[ N' = (-1 + 5, -1 - 3) = (4, -4) \]

4. For point \( P(-3, -1) \): \[ P' = (-3 + 5, -1 - 3) = (2, -4) \]

So, the vertices of the translated square LMNP are:

• \( L'(2, -2) \)
• \( M'(4, -2) \)
• \( N'(4, -4) \)
• \( P'(2, -4) \)

Now let's compare these points with the given squares:

1. Square 1: Points \( (-6, 4), (-6, 6), (-4, 6), (-4, 4) \) (Does not match)
2. Square 2: Points \( (-6, -4), (-6, -6), (-4, -4), (-4, -6) \) (Does not match)
3. Square 3: Points \( (2, 2), (2, 4), (4, 4), (4, 2) \) (Does not match)
4. Square 4: Points \( (2, -2), (4, -2), (4, -4), (2, -4) \) (Matches)

Thus, the image of square LMNP after the translation is Square 4.