MATH 221 Lecture 14

Arun Ram
Department of Mathematics and Statistics
University of Melbourne
Parkville, VIC 3010 Australia
aram@unimelb.edu.au

Last update: 6 August 2012

Graphing Techniques

(a)	Basic Graphs
(b)	Shifting
(c)	Scaling
(d)	Flipping
(e)	Limits
(f)	Asymptotes
(g)	Slopes: Increasing/Decreasing
(h)	Concave Up/Concave down points of Inflection

Basic Graphs

Shifting

Example:$$ Graph ${\left(x-3\right)}^2+{\left(y-2\right)}^2=1$

Scaling

Example:$$ Graph $2y=\sin 3x$

Flipping

Example:$$ Graph $y=-e^{-x}$

Example:$$ Graph $y=\sin \left(\frac{1}{x}\right)$

Notes:
(a)	$y=\sin x$ is the basic graph
(b)	Positive $x$-axis is flipped
(c)	Negative $x$-axis is flipped
(d)	As $x\to \infty ,\sin \left(\frac{1}{x}\right)\to 0^{+}$
(e)	As $x\to -\infty ,\sin \left(\frac{1}{x}\right)\to 0^{-}$
(f)	As $x\to 0^{+},\sin \left(\frac{1}{x}\right)\text{goes between}+1\text{and}-1.$

Example:$$ Graph $y={\sin }^{-1}x$

Asymptotes

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 gets closer and closer to as $x$ gets closer to $a$.

Example:$$ Graph $x^2-y^2=1$

Example:$$ ${\displaystyle f\left(x\right)=\left\{\begin{array}{ccc}\frac{1-\cos x}{x^2},& & \text{if}x\ne 0,\\ 1,& & \text{if}x=0,\end{array}\right.}$

${\displaystyle \underset{x\to 0}{lim}\frac{1-\cos x}{x^2}=\underset{x\to 0}{lim}\frac{1-\left(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\dots \right)}{x^2}}$
${\displaystyle \underset{x\to 0}{lim}\frac{\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\dots }{x^2}=\underset{x\to 0}{lim}\frac{1}{2}-\frac{x^2}{4!}+\frac{x^4}{6!}-\frac{x^6}{8!}+\dots }$
${\displaystyle =\frac{1}{2}-0+0-0+\dots =\frac{1}{2}}$
${\displaystyle \text{So}\underset{x\to 0}{lim}f\left(x\right)=\frac{1}{2}.\text{Since}f\left(0\right)=1,\underset{x\to 0}{lim}f\left(x\right)\ne f\left(0\right).}$
So $f\left(x\right)$ is not continuous at $x=0$.

Notes and References

These are a typed copy of lecture notes given by Arun Ram on October 9, 2000.

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