# Functions and graphs

I have been confusing about the graphs of functions since two or three months. I had searched in internet but could not get something. I want to know how the graphs of polynomials behave. For example, polynomials which have degree greater than two, has special curves. BUT what are the proofs that they are two parabolas. Another simillar questions are for graphs of trigonometric ratios like sin x, cos x,... . And What is the proof of graph of sin 1/x. What is the proof of graphs of exponential and logarithmic functions. In the scientific calculator, we can get graphs of my above questions. Also tell me what will be the graph of $x^{-1}$. Actually my question is what is the proof of that the certain function has a certain graph. Please Also give me the name a book which can solve my above questions. I really needed the book of this subject..

Quite simply, the graph of a function $f:\mathbb{R}\to\mathbb{R}$ is merely the set $$\{(x,f(x)) : x\in \mathbb{R}\}$$ If you want to understand how the graph of a specific function "behaves", you will need to learn some calculus.
For instance, assuming you function is differentiable, then the graph will be "increasing" (ie. as you draw the curve from left to right, the curve goes upward) at a point, if $f' > 0$ at that point.
Similarly, the second derivative $f''$ tells you about the concavity of a graph. For instance, if $f'' > 0$, then the graph will look like a "cup", and if $f'' < 0$, the graph will look like a "cap" around that point.
• There's quite a few : Stewart's $Calculus : Early Transcendentals$ is full of examples. Apostol's $Calculus : Vol 1$ is good for the theory. – Prahlad Vaidyanathan Sep 15 '13 at 16:37