## MATH 221 Lecture 14

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

## Shifting

Example:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}{\left(x-3\right)}^{2}+{\left(y-2\right)}^{2}=1$

## Scaling

Example:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}2y=\mathrm{sin}3x$

## Flipping

Example:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}y=-{e}^{-x}$

Example:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}y=\mathrm{sin}\left(\frac{1}{x}\right)$

 Notes: (a) $y=\mathrm{sin}x\phantom{\rule{0.5em}{0ex}}$ is the basic graph (b) Positive $x$-axis is flipped (c) Negative $x$-axis is flipped (d) As $x\to \infty ,\phantom{\rule{0.5em}{0ex}}\mathrm{sin}\left(\frac{1}{x}\right)\to {0}^{+}$ (e) As $x\to -\infty ,\phantom{\rule{0.5em}{0ex}}\mathrm{sin}\left(\frac{1}{x}\right)\to {0}^{-}$ (f) As $x\to {0}^{+},\phantom{\rule{0.5em}{0ex}}\mathrm{sin}\left(\frac{1}{x}\right)\phantom{\rule{0.5em}{0ex}}\text{goes between}\phantom{\rule{0.5em}{0ex}}+1\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}-1.$

Example:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}y={\mathrm{sin}}^{-1}x$

## 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:$\phantom{\rule{1em}{0ex}}$ Graph $\phantom{\rule{0.5em}{0ex}}{x}^{2}-{y}^{2}=1$

Example:$\phantom{\rule{1em}{0ex}}$ $f\left(x\right)=\left\{\begin{array}{ccc}\frac{1-\mathrm{cos}x}{{x}^{2}},& & \text{if}\phantom{\rule{0.5em}{0ex}}x\ne 0,\\ 1,& & \text{if}\phantom{\rule{0.5em}{0ex}}x=0,\end{array}\right\$

$\underset{x\to 0}{lim}\frac{1-\mathrm{cos}x}{{x}^{2}}=\underset{x\to 0}{lim}\frac{1-\left(1-\frac{{x}^{2}}{2!}+\frac{{x}^{4}}{4!}-\frac{{x}^{6}}{6!}+\dots \right)}{{x}^{2}}$
$\underset{x\to 0}{lim}\frac{\frac{{x}^{2}}{2!}-\frac{{x}^{4}}{4!}+\frac{{x}^{6}}{6!}-\dots }{{x}^{2}}=\underset{x\to 0}{lim}\frac{1}{2}-\frac{{x}^{2}}{4!}+\frac{{x}^{4}}{6!}-\frac{{x}^{6}}{8!}+\dots$
$\phantom{\rule{9.4em}{0ex}}=\frac{1}{2}-0+0-0+\dots =\frac{1}{2}$
$\text{So}\phantom{\rule{0.5em}{0ex}}\underset{x\to 0}{lim}f\left(x\right)=\frac{1}{2}.\phantom{\rule{1em}{0ex}}\text{Since}\phantom{\rule{0.5em}{0ex}}f\left(0\right)=1,\phantom{\rule{0.5em}{0ex}}\underset{x\to 0}{lim}f\left(x\right)\ne f\left(0\right).$
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.