Let's analyze the mathematical function shown in the image.

The function given is:

$f(n) = \left( \frac{\sqrt{3}}{2} + \frac{1}{2} i \right)^n$

This is a complex number raised to the power $n$. The term inside the parentheses has both a real part $\frac{\sqrt{3}}{2}$ and an imaginary part $\frac{1}{2}i$. Let's explore what happens to the function’s graph as $n$ increases.

Detailed Explanation

1. The Complex Base
   The number inside the parentheses, $\frac{\sqrt{3}}{2} + \frac{1}{2}i$, is in the form $a + bi$, where $a = \frac{\sqrt{3}}{2}$ and $b = \frac{1}{2}$. This corresponds to a point in the complex plane.

   • Polar form:
     The complex number’s magnitude (modulus) is:

     $|z| = \sqrt{ \left(\frac{\sqrt{3}}{2}\right)^2 + \left(\frac{1}{2}\right)^2 } = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1$

     The argument (angle with the positive real axis) is:

     $\theta = \tan^{-1}\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right) = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) = \frac{\pi}{6}$

2. Effect of Raising to Power $n$
   In polar form, raising a complex number $z = r e^{i\theta}$ to the power $n$ results in:

   $z^n = r^n e^{in\theta}$

   Since $r = 1$, the magnitude stays 1 for all $n$. Thus, only the angle changes as $n$ increases. Specifically:

   $\text{New angle} = n \cdot \frac{\pi}{6}$

   This means each time $n$ increases by 1, the point rotates by $\frac{\pi}{6}$ radians (or 30 degrees) counterclockwise on the unit circle.

3. Graph Description
   As shown in the provided images:

   • The blue points mark the position of $f(n)$ on the unit circle for increasing values of $n$.
   • For $n = 1$, the point is at an angle $\frac{\pi}{6}$.
   • For $n = 2$, it moves to $2 \cdot \frac{\pi}{6} = \frac{\pi}{3}$, and so on.
   • As $n$ grows, the point continues to rotate around the circle in steps of 30 degrees.

Conclusion

The function $f(n) = \left(\frac{\sqrt{3}}{2} + \frac{1}{2} i\right)^n$ traces points along the unit circle, rotating counterclockwise by $30^\circ$ for each increment of $n$. The red arrow likely shows the direction of rotation, and the blue points mark the positions after each step.

Do you need any further details or explanations?

Here are five related questions to deepen your understanding:

1. What is the difference between polar and Cartesian representations of complex numbers?
2. How do you convert between polar form and rectangular form of a complex number?
3. What happens if the magnitude $r$ were not 1 in this case?
4. Can you generalize how rotations work for other angles in complex numbers?
5. How does complex multiplication affect the magnitude and angle of the product?

Tip:
Always try expressing complex functions in polar form when dealing with powers, as it simplifies understanding rotations and magnitudes!