Chapter 8.2, Problem 68E

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 vectors have opposite directions.

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.