*Section
4-4:Periodic
Functions - Stretching and Translating*
* *
*A function f is
periodic
if there is a positive number p such that:*
*f(x + p) = f(x)*
*for all x in the
domain
of f.*
*The definition
means
that the y values will repeat over some p value called the fundamental
period of the function. Look at the graph:*
*The graph starts at
0,
goes up to 2, back down to 0, down to -2 and back to 0. At this
point
the graph starts repeating. Look at the x value to find the
period
length. The period length p = 4, because it takes 4 units for
this
graph to repeat the y values. You can tell where the graph
will be at larger values by knowing the repeat. f(0) = 0, f(1) =
2, f(2) = 0, f(3) = -2 and f(4) = 0. What is f(21)? Divide
21 by 4 and use the remainder. The remainder is 1. Thus
f(21)
= f(1) = 2*
*What is
f(82)?
Divide 82 by 4. The remainder is 2. Thus f(82) = f(2) = 0.*
* *
*If a periodic graph
has
a maximum value M and a minimum value m, then the amplitude A of the
function
is:*
*A = (M - m)/2*
*The amplitude of
the
above graph is:*
*A = (2 -(-2))/2 =
4/2
= 2*

__Stretching a
graph
vertically__
* *
*The graph of y =
cf(x)
where c is a positive number not equal to 1, is obtained by vertically
stretching or shrinking the graph of y = f(x).*
*Let f(x) be the
following
graph:*
*Now compare these
two
graphs to the green one above.*
*The purple
graph doubled the green
graph. Notice, all that changed was the high and low points of
the
graph. In other words, stretched vertically. Look at the red
graph. It is the green
graph multiplied by 1/2. All that has changed is the high and low
points. In other words, shrunk vertically. The period
length
has not changed. In all three graphs, the period length is 3.*

__Horizontally
stretching
or shrinking__
* *
*The graph of y =
f(cx)
where c is positive and not equal to one is obtained by horizontally
stretching
or shrinking the graph of y = f(x). If c
> 1 it is a horizontal shrink. If
0
< c < 1, it is a horizontal stretch.*
*Here is y = f(x):*
*Watch the effect of
multiplying
the x value by 2.*
* Notice the
amplitude
didn't change. The graph high and low points are the same.
But look at the **purple
graph. Notice it has gone through two complete cycles by the time
the green
graph
has gone through one cycle. It is like compressing a spring.*
*Now watch the
effect
of multiplying by 1/2.*
*The effect this
time
is to stretch the graph. Look at the red
graph. At x = 3, the red
graph is only half way through the cycle. It takes 6 units for
the
red graph to
repeat instead of three for the green
graph. It is like pulling out on a spring.*

__Summary of above__
** **
**If a periodic function
f
has a period p and amplitude A, then:**
** **
**
y = cf(x) has period p and amplitude cA**
** **
**
y = f(cx) has period p/c and amplitude A**

__Translating
Graphs__
* *
*A translation is
simply
moving the exact same graph to another location. The size and
shape
does not change from the original graph, only the placement of the
graph
changes. Your knowledge of basic graphs is very helpful when
doing
translations. Here is how to translate:*
*y - k = f(x - h) is
obtained
by shifting the graph of y = f(x), k units up/down and h units
right/left.*
*You already know
what
y = x*^{2} is. How does y - 2 = (x - 1)^{2}
compare?
*Notice the green
graph is the same size and shape of the blue
graph. It is shifted one unit right and two units up.*
* *
*Now graph y + 1 = |
x
+ 2|. This depends on you knowing that the absolute value graph
is
a v-shaped graph. So this is a translation of y = |x|*
*The green
graph is the graph of the function we want. It is a translation
of
the blue
graph
moved one unit down and 2 units left. Notice in this problem and
the last problem what causes the graph to be shifted right vs. left and
up vs. down.*

__Combining
reflections
and translations__
* *
*When combining
reflections
and translations, remember to reflect first then translate.
Failure
to work the problem in this order may result in the wrong answer.*
*Graph the function
y
- 1 = -|x + 1|*
*The basic graph is
y
= |x|, a v-shaped graph. The negative sign in front makes this a
reflection about the x-axis. Do this first. Then translate
the result by moving the graph up one and one to the left. Our
answer
is in purple.*

*Remember to make a
copy
of the chart on page 142 in your notebook. You must know that
chart!!
It helps a great deal in future chapters!!*

Let's shift into the next
section:
Let's reflect back on the
previous
section: