MATH 221

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

Last update: 13 August 2014

Lecture 14

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

Graph ${(x-3)}^2+{(y-2)}^2=1\text{.}$ $$\begin{array}{c} x 1 2 3 y 1 2 (3,2) 1 \end{array}$$ Notes:

(a)	$x^2+y^2=1$ is a basic circle of radius 1.
(b)	Center is shifted by 3 to the right in the $x$ direction, $2$ upwards in the $y$ direction.

Scaling

Graph $2y=\text{sin}\hspace{0.17em}3x\text{.}$ $$\begin{array}{c} y 1 -1 x -π -5π6 -3π3 -π2 -π3 -π6 π6 π3 π2 2π3 5π6 π \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	The $x$ axis is scaled (squished) by $3\text{.}$
(c)	The $y$ axis is scaled by $2\text{.}$

Flipping

Graph $y=-e^{-x}\text{.}$ $$\begin{array}{c} x y -1 \end{array}$$ Notes:

(a)	$y=e^x$ is the basic graph.
(b)	$y=-e^{-x}$ is the same as $-y=e^{-x}\text{.}$
(c)	The $x$ axis is flipped.
(d)	The $y$ axis is flipped.

Graph $y=\text{sin}\left(\frac{1}{x}\right)\text{.}$ $$\begin{array}{c} x y 1 -1 -1π -12π 12π 1π 2π \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	Positive $x$ axis is flipped.
(c)	Negative $x$ axis is flipped.
(d)	As $x\to \infty ,$ $\text{sin}\left(\frac{1}{x}\right)\to 0^{+}\text{.}$
(e)	As $x\to -\infty ,$ $\text{sin}\left(\frac{1}{x}\right)\to 0^{-}\text{.}$
(f)	As $x\to 0^{+},$ $\text{sin}\left(\frac{1}{x}\right)$ goes between $+1$ and $-1\text{.}$

Graph $y={\text{sin}}^{-1}x\text{.}$ $$\begin{array}{c} x -1 1 y 3π2 π π2 -π2 -π -3π2 \end{array}$$ Notes:

(a)	$y=\text{sin}\hspace{0.17em}x$ is the basic graph.
(b)	$y={\text{sin}}^{-1}x$ is the same as $\text{sin}\hspace{0.17em}y=x$ so $x$ and $y$ axis are switched from $y=\text{sin}\hspace{0.17em}x$ graph.

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 $a\text{.}$

Graph $x^2-y^2=1\text{.}$ $$\begin{array}{c} x y 1 -1 y=x y=-x \end{array}$$ Notes:

(a)	If $y=0$ then $x=\pm 1\text{.}$
(b)	$x^2-y^2=1$ is the same as $1-\frac{y^2}{x^2}=\frac{1}{x^2}\text{.}$ As $x\to \infty$ this becomes $1-{\left(\frac{y}{x}\right)}^2=0\text{.}$ So, as $x\to \infty$ $y^2=x^2\text{.}$ So $y=\pm x\text{.}$ So $y=x$ is an asymptote as $x\to \infty \text{.}$ $y=-x$ is also an asymptote as $x\to \infty \text{.}$

$$f\left(x\right)=\{\begin{array}{cc}\frac{1-\text{cos}\hspace{0.17em}x}{x^2},& \text{if}\hspace{0.17em}x\ne 0,\\ 1,& \text{if}\hspace{0.17em}x=0\text{.}\end{array}$$ $$\begin{array}{ccc}{\displaystyle \underset{x\to 0}{\text{lim}}\frac{1-\text{cos}\hspace{0.17em}x}{x^2}}& =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{1-(1-\frac{x^2}{2!}+\frac{x^4}{4!}-\frac{x^6}{6!}+\cdots )}{x^2}}\\ & =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{\frac{x^2}{2!}-\frac{x^4}{4!}+\frac{x^6}{6!}-\cdots }{x^2}}\\ & =& {\displaystyle \underset{x\to 0}{\text{lim}}\frac{1}{2}-\frac{x^2}{4!}+\frac{x^4}{6!}-\frac{x^6}{8!}+\cdots }\\ & =& {\displaystyle \frac{1}{2}-0+0-0+\cdots =\frac{1}{2}\text{.}}\end{array}$$ So $\underset{x\to 0}{\text{lim}}f\left(x\right)=\frac{1}{2}\text{.}$ Since $f\left(0\right)=1,$ $\underset{x\to 0}{\text{lim}}f\left(x\right)\ne f\left(0\right)\text{.}$ So $f\left(x\right)$ is not continuous at $x=0\text{.}$ $$\begin{array}{c} y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=cos⁡x y 1 -1 x -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=-cos⁡x \\ x y 2 1 -3π -5π2 -2π -3π2 -π -π2 π2 π 3π2 2π 5π2 3π y=1-cos⁡x x y -1 1 1 y=1x2 \\ x y 12 1 y=2x2 y = f(x) = { 1-cos xx2, x≠0, 1, x=0. \end{array}$$ Notes:

(a)	As $x\to 0,$ $\frac{1-\text{cos}\hspace{0.17em}x}{x^2}\to \frac{1}{2}\text{.}$
(b)	As $f\left(0\right)=1\text{.}$
(c)	At the peaks of $1-\text{cos}\hspace{0.17em}x,$ $\frac{1-\text{cos}\hspace{0.17em}x}{x^2}=\frac{2}{x^2}\text{.}$

Notes and References

These are a typed copy of Lecture 14 from a series of handwritten lecture notes for the class MATH 221 given on October 9, 2000.

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