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 = x2 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: