Question

Consider the transformation W=(1+i)z+3-4i. Consider the rectangle ABCD with vertices: A:(1+i); B:(-1+i) ; C:(-1-i), D:(1-i). Do the following

a. Find the image A'B'C'D' of this rectangle under this transformation

b. Plot both rectangles on graph

c. How do the areas of the two rectangles compare?

d. Describe the transformation geometrically?

e. What are the fixed points of the transformation?

Answer 1

transformation W=(1+i)z+3-4i.

A:(1+i); B:(-1+i) ; C:(-1-i), D:(1-i).

a) Find the image A'B'C'D' of this rectangle under this transformation

image A'B'C'D'

A' = (1+i)*(1+i)+3-4i = 3-2i

B' = (1+i)*(-1+i)+3-4i =1-4i

C' = (1+i)*(-1-i)+3-4i = 3 - 6i

D' = (1+i)*(1-i)+3-4i =5-4i

b)Plot both rectangles on graph

c) How do the areas of the two rectangles compare?

Area ABCD = |AB|*|BC| = 2*2 = 4

Area A'B'C'D' = |A'B'|*|B'C'|

Area A'B'C'D' = 2 * Area ABCD

d) Describe the transformation geometrically?

Transformation rotates the rectangle by ,in x and y direction both by units and then translates in x direction by 3 units and in y direction by -4 units

e) What are the fixed points of the transformation?

Let a+ib be the fixed point

then (1+i)*(a+ib)+3-4i = a+ib

(a-b) + i(a+b)+3-4i = a+ib

(a-b+3)+i(a+b-4) = a+ib

a-b+3 = a

a+b-4 = b

a = 4 and b=3

fixed point = 4+3i