The graph of the Julia set in Figure 11 appears to be symmetric with respect to both the x-axis and the y-axis. Complete the following to show that this is true.
(a) Show that complex conjugates have the same absolute value.
(b) Compute Z12 — 1 and z22 - 1, where Z1 = a + bi and Z2 = a - bi.
(c) Discuss why if (a, b) is in the Julia set, then so is (a, - b).
(d) Conclude that the graph of the Julia set must be symmetric with respect to the x-axis.
(e) Using a similar argument, show that the Julia set must also be symmetric with respect to the y-axis.
Concept Check Identify the geometric condition (A, B, or C) that implies the situation.
A. The corresponding
B. The terminal points of the vectors corresponding to a + bi and c + di lie on a horizontal line.
C. The corresponding vectors have the same direction.
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