When a point is dilated by a scale factor of k, the distance between the point and the center of dilation is multiplied by k. The point is moved closer to the center of dilation if k is less than 1, and further away from the center of dilation if k is greater than 1.
In this case, the point (-4,8) is dilated by a scale factor of 1/4 centered at the origin. Since the scale factor is less than 1, the point will be moved closer to the origin. To find the new coordinates, we multiply the distance between the point and the origin by 1/4. The distance between (-4,8) and the origin is:
sqrt((-4)^2 + 8^2) = sqrt(80)
So, the new distance will be:
1/4 * sqrt(80) = sqrt(80)/4 = 2sqrt(5)
To find the new x-coordinate, we multiply the x-coordinate of the point by 1/4:
-4 * 1/4 = -1
To find the new y-coordinate, we multiply the y-coordinate of the point by 1/4:
8 * 1/4 = 2
Therefore, the image of (-4,8) after a dilation by a scale factor of 1/4 centered at the origin is (-1,2).