To identify the rule that describes a dilation with a scale factor of 4 and the center of dilation at the origin, we need to understand what dilation means mathematically. Dilation is a transformation that stretches or shrinks points from a specific center by a specific scale factor.
1. Center of Dilation at the Origin (0,0):
When the center of dilation is at the origin, each point (x, y) is mapped to a new point (x', y') such that both the x and y coordinates are multiplied by the scale factor.
2. Scale Factor:
The scale factor is the ratio by which the coordinates of a point are multiplied. In this case, the scale factor is 4, meaning each coordinate is multiplied by 4.
Using these properties, we apply the dilation:
- For a point (x, y), after dilation by a scale factor of 4 with the origin as the center, the new coordinates (x', y') would be:
- [tex]\( x' = 4x \)[/tex]
- [tex]\( y' = 4y \)[/tex]
Therefore, the rule that describes this transformation is:
[tex]\[ (x, y) \rightarrow (4x, 4y) \][/tex]
Examining the options given:
- A. [tex]\((x, y) \rightarrow (-4x, -4y)\)[/tex] - This rule represents a dilation by a scale factor of -4.
- B. [tex]\((x, y) \rightarrow (4x, 4y)\)[/tex] - This rule represents a dilation by a scale factor of 4.
- C. [tex]\((x, y) \rightarrow (x+4, y+4)\)[/tex] - This rule represents a translation, not a dilation.
- D. [tex]\((x, y) \rightarrow \left( \frac{1}{4}x, \frac{1}{4}y \right)\)[/tex] - This rule represents a dilation by a scale factor of [tex]\(\frac{1}{4}\)[/tex].
Thus, the correct rule is:
[tex]\[ \boxed{(x, y) \rightarrow (4x, 4y)} \][/tex]
