The function $f(n) = \left( \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i \right)^n$ represents a sequence of complex numbers that lies on the unit circle in the complex plane. Let me break this down step by step:

1. The term $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i$ is equivalent to $e^{i\pi/4}$ (a point on the unit circle with an angle of $\pi/4$ radians or 45°).
2. Raising $e^{i\pi/4}$ to the power $n$ gives $e^{in\pi/4}$, which represents a point on the unit circle rotated by $n \cdot \pi/4$ radians.
3. As $n$ increases, the angle of rotation increases by $\pi/4$ for each step, creating a sequence of evenly spaced points around the unit circle.

Analysis of the graphs:

• Graph 1 (Left): It shows a unit circle with 4 points, which corresponds to $n = 4$, creating quarter-circle rotations.
• Graph 2 (Middle): It shows a unit circle with 8 points, consistent with $n = 8$, reflecting eighth-circle rotations.
• Graph 3 (Right): It shows a unit circle with 16 points, matching $n = 16$, showing sixteenth-circle rotations.

Since the problem starts from $n = 1$ and increases $n$, the graph on the right (Graph 3) represents the correct pattern as $n$ grows larger. This is because the number of points doubles with each increase in $n$, leading to finer division of the unit circle.

Would you like further clarification? Here are some follow-up questions:

1. How does the exponential form $e^{i\theta}$ relate to trigonometric functions?
2. What happens to the graph if $\frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} i$ is replaced with another complex number?
3. Why does the pattern form a symmetric distribution on the unit circle?
4. What is the relationship between the power $n$ and the number of points on the graph?
5. How could this concept be extended to visualize other complex transformations?

Tip: When visualizing unit circle behavior, remember that the argument (angle) of a complex number controls the rotation, while the magnitude controls the scaling!