Functions - Stretching and Translating
A function f is
if there is a positive number p such that:
f(x + p) = f(x)
for all x in the
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
goes up to 2, back down to 0, down to -2 and back to 0. At this
the graph starts repeating. Look at the x value to find the
length. The period length p = 4, because it takes 4 units for
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(1) = 2
Divide 82 by 4. The remainder is 2. Thus f(82) = f(2) = 0.
If a periodic graph
a maximum value M and a minimum value m, then the amplitude A of the
A = (M - m)/2
The amplitude of
above graph is:
A = (2 -(-2))/2 =
The graph of y =
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
Now compare these
graphs to the green one above.
graph doubled the green
graph. Notice, all that changed was the high and low points of
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
has not changed. In all three graphs, the period length is 3.
The graph of y =
where c is positive and not equal to one is obtained by horizontally
or shrinking the graph of y = f(x). If c
> 1 it is a horizontal shrink. If
< c < 1, it is a horizontal stretch.
Here is y = f(x):
Watch the effect of
the x value by 2.
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
has gone through one cycle. It is like compressing a spring.
Now watch the
of multiplying by 1/2.
The effect this
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
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
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
A translation is
moving the exact same graph to another location. The size and
does not change from the original graph, only the placement of the
changes. Your knowledge of basic graphs is very helpful when
translations. Here is how to translate:
y - k = f(x - h) is
by shifting the graph of y = f(x), k units up/down and h units
You already know
y = x2 is. How does y - 2 = (x - 1)2
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 = |
+ 2|. This depends on you knowing that the absolute value graph
a v-shaped graph. So this is a translation of y = |x|
graph is the graph of the function we want. It is a translation
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.
and translations, remember to reflect first then translate.
to work the problem in this order may result in the wrong answer.
Graph the function
- 1 = -|x + 1|
The basic graph is
= |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
is in purple.
Remember to make a
of the chart on page 142 in your notebook. You must know that
It helps a great deal in future chapters!!
Let's shift into the next
Let's reflect back on the