Question

which of the following transformations is a dilation?

○ (x, y) → (x - 4, y)

○ (x, y) → (x + 4, y + 4)

○ (x, y) → (4x, y)

○ (x, y) → (4x, 4y)

Explanation:

Step1: Recall Dilation Definition

Dilation transforms a figure by scaling (multiplying) coordinates by a scale factor, keeping the center (usually origin) fixed. Translations (adding/subtracting constants) are not dilations.

Step2: Analyze Each Option

• Option 1: \((x, y) \to (x - 4, y)\) is a horizontal translation (subtracting 4 from \(x\)), not dilation.
• Option 2: \((x, y) \to (x + 4, y + 4)\) is a translation (adding 4 to \(x\) and \(y\)), not dilation.
• Option 3: \((x, y) \to (4x, y)\) scales only \(x\)-coordinate, not uniform scaling (dilation requires scaling both \(x\) and \(y\) by the same factor for center at origin, or proportional scaling).
• Option 4: \((x, y) \to (4x, 4y)\) scales both \(x\) and \(y\) by factor 4, which is a dilation (uniform scaling from origin).

Answer:

\((x, y) \to (4x, 4y)\) (the fourth option: \(\boldsymbol{(x, y) \to (4x, 4y)}\))