How Do Composite Transformations Affect Coordinates?

[golb0016]

1.) If (-4,8) is the image under the composite transformation T_h,3 x T_-2,k(-3,0), what are the coordinates of the image of (2,-1) under the same composite transformation?

The Attempt at a Solution

I'm lost on this one.

 

[HallsofIvy]

I assume that "T_h,3" is translation by "h" in the x direction and "3" in the y direction- that is, that T_h,3(x, y)= (x+h, y+3)- and that "T_{-2, k}" is translation by "-2" in the x direction and "k" in the y direction- that is, that T_-2,k(x,y)= (x-2, y+ k).

If that is so then T_h,3 x T_-2,k(-3, 0)= T_h,3(-3-2, 0+k)= T_h,3(-5,k)= (-5+h, 3+k)= (-4,8). That is, -5+h= -4 and 3+ k= 8. Now, what are h and k? Once you know that, finding T_h,3x T_-2,k(2, -1) should be easy.

 

[Architeuthis Dux]

A composite transformation is a combination of two or more transformations, usually represented by the notation "Th x T", where Th represents one transformation and T represents another. In this case, the composite transformation is Th,3 x T-2,k(-3,0). This means that the original point (-4,8) is first translated by 3 units in the horizontal direction (Th,3), and then reflected over the line x=-2 and stretched by a factor of k (T-2,k).

To find the image of (2,-1) under the same composite transformation, we can follow the same steps. First, we translate (2,-1) by 3 units in the horizontal direction, which gives us (5,-1). Then, we reflect over the line x=-2, which gives us (-6,-1). Finally, we stretch by a factor of k, which gives us the final coordinates of the image as (-6k,-k).

Therefore, the coordinates of the image of (2,-1) under the composite transformation Th,3 x T-2,k(-3,0) would be (-6k,-k).