Download Transformations, Infinity, and Graphing

There are two kinds of limits involving the idea of infinity…
1) Where the limit DNE or is +/- infinity
2) Where there is a limit as the value approaches +/- infinity
Type 1 Example:
Note: If the

problem
does not
specify a
right or left
hand limit,
you must
check both
to make
sure they
equal each
other.
• Our friend Patrick will show us an example of how to
solve a limit problem at infinity without the use of
shortcuts
• The important component for limits at infinity (with
rational polynomials) is to divide by the HIGHEST
POWER
• However, there are a few shortcuts…
1) Degree of Numerator=Degree of Denominator
Limit (as x approaches infinity) =
Ratio of Leading Coefficients of Highest Degree
2) Degree of Numerator<Degree of Denominator
Limit (as x approaches infinity)=
3) Degree of Numerator>Degree of Denominator
Limit (as x approaches infinity)=
An asymptote of a graph is a line where the function does
not ever cross
A horizontal asymptote looks like this:
A vertical asymptote looks like this:
Horizontal and vertical asymptotes
are prevalent in limit problems….
For example: This graph could be used with
problem that asks…
what is the limit as x approaches 0??
The answer?? DNE
Why??

YAxis
Xscale
-2
YOrigin intercept
-1
-1
Y-scale
Quadrant 1: (+,+)
Quadrant 2: (-,+)
Quadrant 3: (-,-)
Quadrant 4: (+,-)
X-axis
Xintercept
-2
Point P corresponds to the pair (a,b).
Fun Fact:
Rene
Decartes
invented the
Cartesian
coordinates.
Types of Graphs and Real Life
Example
Coach
to run
y=x^3
y=x^2 told Johnny
a route: 2 steps forward,
cut right 4 steps, up 3
steps, and to the right
again 6 steps. Make a
table of the (x,y) pairs
that are on the path of
the
football, if ity=IxI
is thrown
y=√(x)
to Johnny at the end of
his route.
x
y
0
0
2
1
4
2
6
3
8
4
10
5

Vertical Compression by 1/2
1/2f(x)

Horizontal Compression by
1/2
f(2x)

Vertical Stretch by 2
2f(x)

Horizontal Stretch by 2
f([1/2]x)

Reflection over x-axis
-f(x)

Reflection over y-axis
f(-x)

Shift right 1
f(x-1)

Shift left 1
f(x+1)

Shift up 1
f(x)+1

Shift down 1
f(x)-1
f(x)
-2f(x)
f(2x)
f(-2x+8)

Real Life Example:
Timmy is mapping a route
from his house to his school,
which is across the field
behind his house. He draws
a line following his route,
represented by the function
f(x), but he is moving two
houses down the street.
What should his new function
be to represent his route?
Red is school, blue is the old
house, yellow is the new
house
Answer: f(2x)

The most important thing
we learned is how to solve
limits involving infinity. First,
we learned the longer way
and the reasoning behind
the math. Then, we learned
a shortcut to speed up the
process.
Textbook
http://www.sosmath.com/calculus/li
mcon/limcon04/limcon04.html
http://www.mathsisfun.com/calculus
/limits-infinity.html
http://tutorial.math.lamar.edu/Classe
s/CalcI/LimitsAtInfinityI.aspx
-Patrick JMT youtube channel
Sources:
http://www.mathsisfun.com/definitions/quadrantgraph-.html
http://accelerateu.org/resourceguides/math/m8_
38.gif
http://jwilson.coe.uga.edu/EMAT6680Fa07/Gilbert/
Assignment%202/Gayle-2_files/image004.png
http://upload.wikimedia.org/wikibooks/en/archive
/2/26/20061012185938!Y%3DX%5E3.svg
http://jmckennonmth212s09.files.wordpress.com/2
009/03/sqroot-of-x1.png%3Fw%3D354%26h%3D379
http://media1.shmoop.com/images/algebraii/alg2_ch2_narr_graphik_31.png
http://kartoweb.itc.nl/geometrics/Bitmaps/2D%20
Cartesian%20coordinate%20system.gif
http://mathworld.wolfram.com/images/epsgif/Interval_1000.gif
http://accelerateu.org/resourceguides/math/m8_
38.gif