What is the image of the point A(3, -4) under the dilation by factor 2 with center of dilation at the origin followed by translation by the vector (-1, 1)

Sure, let's solve this step by step.

Step 1: Dilation A dilation is a transformation that changes the size of a figure without altering its shape. It is defined by its center and a scale factor. In this case, the center of dilation is the origin (0,0) and the scale factor is 2.

The formula for dilation is (x', y') = (kx, ky), where (x', y') are the coordinates of the image, (x, y) are the coordinates of the original point, and k is the scale factor.

So, for the point A(3, -4), after dilation, the new coordinates A'(x', y') will be:

x' = 2 * 3 = 6 y' = 2 * -4 = -8

So, the image of the point A(3, -4) after dilation is A'(6, -8).

Step 2: Translation A translation is a transformation that moves every point of a figure the same distance in the same direction. It is defined by a vector (-1, 1) in this case.

The formula for translation is (x", y") = (x' + a, y' + b), where (x", y") are the coordinates of the image after translation, (x', y') are the coordinates of the point after dilation, and (a, b) is the vector of translation.

So, for the point A'(6, -8), after translation, the new coordinates A"(x", y") will be:

x" = 6 + (-1) = 5 y" = -8 + 1 = -7

So, the image of the point A(3, -4) under the dilation by factor 2 with center of dilation at the origin followed by translation by the vector (-1, 1) is A"(5, -7).