Graph I : Converting Functions to Graph

Graphs (2 D) are pretty cool, because they show us how a function, no matter how complicated, behaves over an interval. We can see how a function, which looks super complicated on a piece of paper beautifully unfolds itself, into a picture, and it is then, super easy to decipher the meaning of the function.

In this Blog, I will discuss how to convert an equation to a Graph.

Converting an equation to a graph is a very useful skill, because it helps to understand the function you are working with. It is also pretty fun. But, before that, it is essential to Understand the Concept of Maxima and Minima. 

The maxima is a point on a curve, such that it has a higher value than the points near it in a close range.

The minima is a point on a curve, such that it has a lower value than the points near it in a close range.

Given a function y = f(x), the value of x, such that it's derivative at that point is 0, gives either a maxima, or a minima of the function. Though, there are some cases where neither maxima, nor minima are found by that method. It is actually easy to find if such an x is the maxima or the minima.

Eg : - f(x) = sin x

          f '(x) = cosx

          cos x = 0 at x = π/2, 3π/2, ... (2n - 1)π/2 where n is an integer.

          Therefore, maxima or minima, are at such points. To figure out which is maxima or minima,

         Draw a table, as below, and mark the sign of f '(x) in between two consecutive roots, by taking any value in the range and marking its sign.

               -π/2       π/2          3π/2          5π/2        7π/2

x	0	π	2π	3π
f '(x)	+	-	+	-

If you look at π/2, the slope goes from + to -, on the left part of the graph the values of f(x) increases to reach the value of of f(π/2), and after it reaches that value, the value starts to decrease, as slope is negative. Thus, f(π/2) is a maximum. Therefore, if the f '(x) goes from + to -, that value of f(x) is a maximum. On the converse, from - to +, f(x) is the minimum.

Though, that might not be the maximum value of the function, but in a close range, it is. 

Sometimes, it might not even go from - to +, or + to -, like f(x) = .

f(x) = 

f '(x) = 3 = 0 at x = 0

Before 0, f '(x) is +, after, it is +, so the slope is constantly increasing, but that does not mean it does not have a maxima or a minima. Actually, for any case, we also need to look at the value of the function at the endpoints of the function.

For f(x) = , f() ->  and f(-) -> - , so the minima and the maxima are -  and  respectively.

Inflection Point is the point where the the slope of a curve changes it's sign. The roots of f "(x), and the points where            f(x) ->  are called Inflection Points.

If f '(x) is increasing, it is called Concave Up, and if it is decreasing, it is called Concave Down, i.e. If      f"(x) > 0 in a region, it is called Concave Up, and if f"(x) < 0, it is Concave Down. Looking at these graphs, it is easy to explain the concept.

                               

Now, for the main topic of the blog, how to draw graphs from an equation :

1) Find all the roots of f(x).

2) Find out all asymptotes, i.e. f(x) ->  .

3) Find all Max-Min (also called Critical Points).

4) Find all Inflection Points.

5) Also see where the function goes at -  and +.

Then, join the points according to their properties. Look at the following example to understand:

This Graph is called Newton's Serpentine.

f(x) = 0, at (0, 0).

f '(x) = 0, at (-1, -2), (1, 2).

f"(x) = 0, at (- , - ), (0, 0), (, ).

                           -           0            

x	-	- 1	1
f"(x)	-	+	-	+

f(x) ->  does not exist for any real x.

f(- ) -> 0

f() -> 0

Here is a rough sketch of the graph, indicating all the points, and basically how they are connected :

To see an example related to asymptotes :

f(x) = 0, at (- , 0), (, 0)

f '(x) = 0, at (- 1, - 2), (1, 2)

f "(x) = 0, at (- , 0), (,- ), (, ), (, 0)

Asymptotes - (,- ), (, )

f(- ) = -

f() = 

It's graph looks like :

Vertical dotted line shows Asymptote.

And, that is it, how to graph any two dimensional function, on a piece of paper.