# Thread: Fourier series and sketching graphs

1. ## Fourier series and sketching graphs

Hi Im new to Fourier Series and have a question about sketching graphs by using the following information:

1). f(x) = 3x - x^2 0 < x < 3
f(x) = f(x + 3)

2) f(x) = 2 sin x 0 < x < pi
f(x) = 0 pi < x 2pi
f(x) = f(x + 2pi)

how do I use this info to sketch the graphs?

2. 1) The first one is just a parabola. Sketch it from 0 to 3 and then just copy/paste it across the neighboring intervals.

2) This is the same. Just sketch one interval and then just copy/paste.

Now if you are having trouble sketching the functions themselves, that's another problem.

3. These are "piecewise" defined functions. (My first reaction was "f(x) is NOT equal to f(x+ 3) for this function", but I see that you mean f(x) is defined by the quadratic for x between 0 and 3 and then is periodic with period 3- it is "continued by periodicity".) Start with the graph of $f(x)= 3x- x^2$ for $0< x< 3$. We can see that $f(x)= x(3- x)$ so the graph contains (0, 0) and (3, 0). Also, by completing the square, f(x)= -(x^2- 3x+ (9/4)- (9/4))= -(x- 3/2)^2+ 9/4 so the vertex of the parabola is at (3/2, 9/4). For x between 0 and 3, that is a parabola which rises from (0, 0) up to (3/2, 9/4) then drops back down to (3, 0).
Now, just repeat exactly that graph between x= 3 and x= 6, x= 6 and x= 9, etc. as well as for x between -3 and 0, between -6 and -3, etc. Your graph should be a succession of little "pieces of parabola", all above the x-axis.

For the second problem, you are expected to know the graph of the "basic" f(x)= sin(x). That is a periodic graph that starts at (0, 0), rises to (pi/2, 1), then down to (pi, 0), (3pi/2, -1) and finally rises up to (2pi, 0), repeating that with period 2pi.

Multiplying the function by 2 just multiplies the y values: 2 sin(x) starts at (0, 0), rises to (pi/2, 2), then back down to (pi, 0). Since you are told that this "piece" is only for x between 0 and pi, that is what you start with. Now you are told that f(x)= 0 for x between pi and 2p so the graph is one "hump" of 2 sin(x) from x= 0 to pi, then just the x-axis from x= pi to 2p. Again, it is "continued by periodicity" so you just copy that graph from x= 2pi to 4pi, -2p to 0, etc.

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# how to sketch fourier series graph

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