Expert: Josh - 7/13/2005

Question
I want to find a website about the simple rules and instructions in drawing graphs easily. Can you help?

Answer
Nassim,

I don't know any more than you do off the top of my head. Best way is to type in keywords like "functions", "graphing technique" etc. using your favorite search engine.

For a start, you know that a function f(x) accepts an input value x, and returns a value which is determined solely by x.

Often, the asymptotic values are of interest.
Consider f(x)=x/(1+x) for example.

(i) consider positive values of x.
When x is close to 0, i.e., x->0+, f(x) is approximately 0.
As x becomes larger and larger, limit of x/(1+x) as x tends toward large positive values, is dominated by "x/x". This is because the term "1" becomes neglible in comparison to a very large x value. So, f(x) approaches 1.

(ii) consider negative values of x.
lim x/(1+x) as x tends toward negative infinity, is also dominated by "x/x". This is because the term "1" becomes neglible in comparison to a very large negative x value. So, f(x) approaches 1.

(i) and (ii) gives you horizontal asymptotes (limiting values).

(iii) Find out places where f(x) is undefined. Specifically, f(x)=x/(1+x) is not defined if the denominator is 0. This happens when (1+x)=0, i.e., when x approaches -1. There are two possibilities.
(a) x approaches -1 from the negative side,
e.g., => -1.03, -1.02, -1.01,...
we have (-1.01)/(1+(-1.01))=-1.01/(-0.1) -> large POSITIVE value (the curves shoots up)
(b) x approaches -1 from the positive side,
e.g., ..., -0.99, -0.98, <=
we have (-0.99)/(1+(-0.9))=-1.01/(0.1) -> large NEGATIVE value (the curves drops down rapidly)

So, you expect the graph to behave like this.

When x-> -infinity, f(x)->1,
[the notation "->" means "approaches"]
It increases as x goes from -infinity towards (but not including) x=-1.
As x-> -1 (from left hand side), f(x)-> +infinity
There is an essential discontinuity at x=-1.
Beyond that, at x=-1+epsilon, where epsilon is an extremely small positive value, we see that f(x) climbs up from -infinity.
At x=0, f(x)=0.
As x increases further, it has a limiting value of f(x)=1, as x -> +infinity.