Sketching a periodic function and Fourier analysis

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SUMMARY

The discussion centers on sketching the periodic function f(x) = x² within the interval -3a < x < a, where the function is defined as periodic with a period of 2a. The user clarifies that a function is periodic if it repeats its values over a specified interval, and in this case, the function must be sketched from -a to a and then repeated in the intervals -3a to -a and a to 3a. The conclusion is that the periodic nature of the function can be established by replicating the graph of f(x) = x² within the defined intervals.

PREREQUISITES
  • Understanding of periodic functions and their properties
  • Familiarity with Fourier coefficients and their calculation
  • Basic knowledge of graphing quadratic functions
  • Concept of function intervals and periodicity
NEXT STEPS
  • Study the concept of Fourier series and how they apply to periodic functions
  • Learn about the graphical representation of periodic functions
  • Explore the implications of function transformations on periodicity
  • Investigate the properties of even and odd functions in relation to periodicity
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Students studying calculus, particularly those focused on Fourier analysis, as well as educators and anyone interested in understanding the graphical representation of periodic functions.

SU403RUNFAST
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Homework Statement


So i have a function f(x)=x^2 that is periodic -a<x<a and need to sketch this function from -3a<x<a. I know how to find the Fourier coefficients though.

Homework Equations


f(x)=x^2 sketch it periodically

The Attempt at a Solution


I know that a function is only periodic if it repeats its value in a given interval, so f(x)=f(x+D) where D is some distance down the x-axis where f(x) has the same value as f(x+D), and I know what a graph of x^2 looks like in my head but it does not repeat... How is this function periodic? Sketching it is the first step of my homework problem. It appears to me that the interval given -a<x<a is supposed to be the interval of which it repeats, so it would repeat -3a<x<a and a<x<3a. Is the problem given to us missing other conditions? A sketch or some ideas is helpful thanks in advance
 
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SU403RUNFAST said:
It appears to me that the interval given -a<x<a is supposed to be the interval of which it repeats,
I think that should be the case. This just means that the period is ##2a##.
 
You plot it from -a to +a and copy that picture 2a to the right, 4a etc.
 
That makes sense, thanks I can complete the question
 

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