To find the coordinates of point \( C' \) after dilating point \( C \) by a scale factor of 2 with the center of dilation at the origin, we first need to identify the coordinates of point \( C \).
From your list, it appears point \( C \) is at the coordinates (0, 1).
When we perform a dilation from the origin (0,0) with a scale factor of 2, we multiply each coordinate by the scale factor:
\[ C' = (x, y) \rightarrow (2 \cdot x, 2 \cdot y) \]
So, for point \( C(0, 1) \):
\[ C' = (2 \cdot 0, 2 \cdot 1) = (0, 2) \]
Thus, the coordinates of point \( C' \) are \( (0, 2) \).
So the answer is:
Point \( C' \) is at \( (0, 2) \).