## Graphing

Last update: 25 July 2012

## Basic graphs

### The Basic Circle

What is the equation of the basic circle?

Distances:

For the moment, call it $r$.

So

$\begin{array}{ccccc}\text{Area of the outer square}& =& \text{Area of the inner square}& +& \text{Area of 4 triangles}\\ {\left(p+q\right)}^{2}& =& {r}^{2}& +& 4·\frac{1}{2}pq\end{array}$

This is the heavily used Pythagorean Theorem.

So what is the equation of the basic circle?

$\begin{array}{ccc}\sqrt{{x}^{2}+{y}^{2}}=1& \text{is}& \text{all points}\phantom{\rule{0.2em}{0ex}}\left(x,y\right)\phantom{\rule{0.2em}{0ex}}\text{that are distance 1 from the origin.}\end{array}$

So ${x}^{2}+{y}^{2}=1$ is the equation of the basic circle.

### The Basic Hyperbola $\phantom{\rule{2em}{0ex}}{x}^{2}-{y}^{2}=1$

If $x$ gets very big $\frac{1}{x}$ gets closer and closer to 0 and the equation gets closer and closer to $1-{\left(\frac{y}{x}\right)}^{2}=0$. This is the same as ${\left(\frac{y}{x}\right)}^{2}=1$, which is the same as . So as $x$ gets very large the equation gets closer and closer to $y=x$ and $y=-x$.

As $x$ gets very negative the basic hyperbola gets closer and closer to $y=x$ and $y=-x$.

An asymptote of a graph $y=f\left(x\right)$ as $x\to a$ is another graph $y=g\left(x\right)$ that the original graph $y=f\left(x\right)$ gets closer and closer to as $x$ gets closer and closer to $a$.

Example:    Graph  $y=\frac{1}{x}$

 $y$ $x$ $-1$ $1$ $-1$ $1$

## Notes and References

These notes are from MATH 272 Lectures 1 and 2 given by Arun Ram on Jan 25 2000, Jan 27 2000.