To determine the rule that describes a dilation with a scale factor of [tex]\(\frac{1}{3}\)[/tex] and the center of dilation at the origin, we need to examine the effect of dilation on the coordinates of any given point.
A dilation transformation scales a point [tex]\((x, y)\)[/tex] by a given factor while keeping the point at the center of dilation fixed. When the center of dilation is the origin [tex]\((0, 0)\)[/tex], the transformation affects both the [tex]\(x\)[/tex]-coordinate and the [tex]\(y\)[/tex]-coordinate equally based on the scale factor.
The mathematical rule for dilation with a scale factor [tex]\(k\)[/tex] centered at the origin is given by:
[tex]\[
(x, y) \rightarrow (k \cdot x, k \cdot y)
\][/tex]
In this specific case, the scale factor [tex]\(k\)[/tex] is [tex]\(\frac{1}{3}\)[/tex]. Therefore, the rule for the dilation transformation is:
[tex]\[
(x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right)
\][/tex]
Let's analyze the given options based on this rule:
- Option A: [tex]\((x, y) \rightarrow \left(\frac{1}{3}x, \frac{1}{3}y\right)\)[/tex]
- This matches our derived rule.
- Option B: [tex]\((x, y) \rightarrow \left(x + \frac{1}{3}, y + \frac{1}{3}\right)\)[/tex]
- This represents a translation, not a dilation.
- Option C: [tex]\((x, y) \rightarrow \left(-\frac{1}{3}x, -\frac{1}{3}y\right)\)[/tex]
- This is a dilation by a scale factor of [tex]\(-\frac{1}{3}\)[/tex], indicating a reflection and a scaling, which does not match our requirement.
Hence, the correct rule that describes a dilation with a scale factor of [tex]\(\frac{1}{3}\)[/tex] and the center of dilation at the origin is:
[tex]\[
\boxed{(x, y) \rightarrow \left(\frac{1}{3} x, \frac{1}{3} y\right)}
\][/tex]
