Shifting and Reflecting

Six Basic Functions
Below are six basic functions:






Memorize the shapes of these functions.

Horizontal Shifting
Consider the graphs
y = ^{}

(x + 0)^{2
}

(x + 1)^{2
}

(x + 2)^{2
}

(x + 3)^{3}
Exercise
Use the list features of a calculator to sketch the graph of
y = 1/[x  {0,1,2,3}]
Rule1: f(x  a) = f(x) shifted a units to the right.
Rule 2: f(x + a) = f(x) shifted a units to the left


Vertical Shifting
Consider the graphs
y =

x^{3
}

x^{3} + 1^{
}

x^{3} + 2^{
}

x^{3} + 3
Exercise
Use the list features of a calculator to sketch the graph of
y =
x^{3}  {0,1,2,3}
Rule 3: f(x ) + a = f(x) shifted a units up.
Rule 4: f(x)  a = f(x) shifted a units down. 

Reflecting About the xaxis
Consider the graphs
y = x^{2} and y = x^{2}
Rule 5: f(x ) = f(x) reflected about the xaxis. 

Reflecting About the yaxis.
Exercise:
Use the calculator to graph
and
Rule 6: f(x ) = f(x) reflected about the yaxis. 

Stretching and Compressing
Exercises
Graph the following
y = {1,2,3,4}x^{3
}
y = {1/2,1/3,1/4,1/5}x^{3}
^{
}
Rule 7: cf(x ) = f(x) (for c > 1) stretched vertically.
Rule 8: cf(x ) = f(x) (c < 1) compressed vertically. 
We will do some examples (including the graph of the winnings for the gambler
and for the casino.
Exercises: Graph the following

y = x^{2}  10

y = sqrt(x  2)

y = x  5 + 3
For an interactive investigation of the shifting rules go
to
http://mathcsjava.emporia.edu/~greenlar/Shifter/Shifter1.html

Increasing and Decreasing Functions
A function is called increasing if as an object moves from
left to right, it is moving upwards along the graph.
If
x < y
then
f(x) < f(y)
A function is called decreasing if as an object moves from
left to right, it is moving downwards along the graph.
If
x < y
then
f(x) > f(y)
Example:
The curve
y = x^{2}
is increasing on (0,
) and decreasing
on (,0)
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