Describe graphs in Fourier series. Hello friends, here I show how to describe graphs in Fourier series. Have a look!!

If you’re looking for more in the graphs of Fourier series, do check in:

**How to draw graphs of functions in Fourier series**

describe graphs_compressed

**Describe graphs in Fourier series**

**Solved examples of how to describe graphs in Fourier series**

Note: None of these examples is mine. I have chosen these from some books. I have also given the due reference at the end of the post.

So here is the first example of how to describe graphs in Fourier series.

**Example 1**

According to Stroud and Booth (2011), “For the following graph give the analytical description of the function drawn.

**Solution**

Now I have to find out the analytical description of this graph.

So this means that I have to describe the graph of the function . And this is one of the basic stuff to learn in Fourier series.

Hence comes my first step.

**Step 1**

As I can see from the graph, the function

So I can say that the function

(1)

Now I’ll describe the function.

**Step 2**

As I can see from the graph, at

And at

Now I’ll find out the equation of the straight line passing through two points

As we all know, the standard equation of a straight line passing through two points

So the equation of the straight line passing through two points

Now I’ll simplify it to get

So this means

Next, I’ll replace

In mathematical term, I can say

(2)

Now I’ll find out the expression for

**Step 3**

As I can see from the graph, the value of

(3)

Therefore I can combine equations (1), (2) and (3) to describe the function

Hence I can conclude that this is the answer to the above-mentioned example.

Now I’ll give another example of how to describe graphs in Fourier series.

**Example 2**

“For the following graph give the analytical description of the function drawn.

**Solution**

Now here again I’ll find out the analytical description of this graph.

So again I’ll describe the function

Hence comes my first step.

**Step 1**

As I can see from the graph, the function

So I can say that the function

(4)

Now I’ll describe the function.

As I can see from the graph, the value of

(5)

Now I’ll find out the expression for

**Step 2**

As I can see from the graph, at

And at

Now I’ll find out the equation of the straight line passing through two points

As we all know, the standard equation of a straight line passing through two points

So the equation of the straight line passing through two points

Now I’ll simplify it to get

So this means

Next, I’ll replace

In mathematical term, I can say

(6)

Now I’ll find out the expression for

**Step 3**

As I can see from the graph, at

And at

Now I’ll find out the equation of the straight line passing through two points

So the equation of the straight line passing through two points

Now I’ll simplify it to get

So this means

Next, I’ll replace

In mathematical term, I can say

(7)

Therefore I can combine equations (4), (5), (6) and (7) to describe the function

Hence I can conclude that this is the answer to the above-mentioned example.

Now I’ll give another example of how to describe graphs in Fourier series.

**Example 3**

“For the following graph give the analytical description of the function drawn.

**Solution**

Now here again I’ll find out the analytical description of this graph.

So again I’ll describe the function

Hence comes my first step.

**Step 1**

As I can see from the graph, the function

Also, I can see that the function

So I’ll start when

At

And at

Now I’ll find out the equation of the straight line passing through two points

So the equation of the straight line passing through two points

Now I’ll simplify it to get

So this means

Next, I’ll replace

In mathematical term, I can say

(8)

Next, I’ll choose the interval between

**Step 2**

As I can see from the graph, the value of

(9)

Now I’ll find out the expression for

**Step 3**

At

And at

Now I’ll find out the equation of the straight line passing through two points

So the equation of the straight line passing through two points

Now I’ll simplify it to get

So this means

Next, I’ll replace