Here is the graph of the function \( f(x) = \frac{1}{12}(x^2 - \pi^2) \) over the interval \([-4\pi, 4\pi]\). The graph shows the periodic nature of the function, repeating every \(2\pi\) with a parabolic shape, reaching its minimum at \(x = 0\) and \(x = \pm 2\pi\).

Would you like to proceed with the Fourier series analysis or have any questions?

### Relative Questions:
1. How do you calculate the period of a periodic function?
2. What is the significance of the vertex of the parabola in this context?
3. How does the periodic nature of this function manifest in its Fourier series?
4. What are the steps to graph a periodic function over multiple intervals?
5. Can the function's symmetry help in simplifying calculations for its Fourier series?
6. How do the critical points of this function influence its Fourier coefficients?
7. What role does the period play in determining the behavior of the Fourier series?
8. How can the Fourier series representation be used to approximate non-periodic functions?

### Tip:
When graphing periodic functions, always ensure that you include multiple periods to fully capture the repeating pattern and its effects.

Explore the graph of the function f(x) = (1/12)(x^2 - π^2) over the interval [-4π, 4π]. This periodic function exhibits a parabolic shape, with minima at x = 0 and x = ±2π. Learn about its Fourier series analysis and periodic nature implications.

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