1. **Stating the problem:** If a point $A(x,y)$ is enlarged at the origin by a scale factor $k$, find the coordinates of the image point $A^\prime$. 2. **Formula used:** When a point is enlarged from the origin by a scale factor $k$, the coordinates of the image point $A^\prime$ are given by: $$A^\prime = (kx, ky)$$ 3. **Explanation:** - The origin is the center of enlargement. - The scale factor $k$ stretches or shrinks the distance of the point from the origin. - If $k > 1$, the point moves away from the origin. - If $0 < k < 1$, the point moves closer to the origin. - If $k$ is negative, the point is also reflected through the origin. 4. **Intermediate work:** Given $A = (x, y)$ and scale factor $k$, multiply each coordinate by $k$: $$x^\prime = kx$$ $$y^\prime = ky$$ 5. **Final answer:** The image point after enlargement is: $$A^\prime = (kx, ky)$$