# Sketching a periodic function and Fourier analysis

• SU403RUNFAST
In summary, the conversation discusses how to sketch a periodic function f(x)=x^2 from -3a<x<a and finding the Fourier coefficients for this function. It is noted that a function is only periodic if it repeats its value in a given interval, and the interval given for this function is -a<x<a. The individuals in the conversation discuss how to plot the function from -a to a and then copy it to the right to get the periodic behavior.
SU403RUNFAST

## 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

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

## 1. What is sketching a periodic function?

Sketching a periodic function involves graphing a function that repeats itself at regular intervals. This is typically done over a specific domain, such as a certain range of values on the x-axis, to show the pattern of the function.

## 2. How do you determine the period of a periodic function?

The period of a periodic function is the smallest interval over which the function repeats itself. It can be determined by finding the distance between two consecutive peaks or troughs on the graph of the function.

## 3. What is Fourier analysis?

Fourier analysis is a mathematical technique used to break down a complex function into simpler components. It involves representing a function as a sum of sine and cosine waves, which allows for a better understanding of the behavior of the function.

## 4. How is Fourier analysis used in sketching a periodic function?

Fourier analysis can be used to identify the different components of a periodic function, such as the amplitude, frequency, and phase. This information can then be used to sketch the function accurately and understand its behavior.

## 5. Are there any limitations to using Fourier analysis in sketching periodic functions?

While Fourier analysis can provide valuable insights into the behavior of a periodic function, it does have limitations. For example, it may not accurately represent functions with sharp corners or discontinuities. Additionally, it may not work well with functions that have infinitely many peaks or troughs.

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