Transcript Lesson 4-4 Stretching and Translating Graphs Various functions ‘repeat’ a set of values.

Lesson 4-4
Stretching and
Translating Graphs
Various functions ‘repeat’ a set of
values.
Various functions repeat a set of
values.
Their graphs will be a
repetition of a basic curve.
Period of the function:
Period of the function:
The length of the x-cycle
that it takes for the curve
to repeat itself.
When p equals the period of the
function (the interval of x-values it
takes for a curve to repeat its cycle)
we can say,
When p equals the period of the
function (the interval of x-values it
takes for a curve to repeat its cycle)
we can say,
f(x+p) = f(x)
for all x in the domain of x.
Example:
Example:
The graph of a periodic function f is shown on
page 139.
Find:
Example:
The graph of a periodic function f is shown on
page 139.
Find:
a) The fundamental period of f.
Example:
The graph of a periodic function f is shown on
page 139.
Find:
a) The fundamental period of f.
If you start at the origin and follow
the graph to the right, the graph
takes 4 units to complete one up
and-down cycle. So, the period is 4.
Example:
The graph of a periodic function f is shown on
page 139.
Find:
b) f(99)
Example:
The graph of a periodic function f is shown on
page 139.
Find:
b) f(99)
If we take x = 99, divide by 4 (the period), we
get 24 with a remainder of 3.
Therefore, we can show:
f(99) = f(4(24) + 3)
= f(3) = - 2
If a periodic function has a
maximum value M and a
Minimum value m, then the
amplitude of a function is given
by:
If a periodic function has a
maximum value M and a
minimum value m, then the
amplitude of a function is given
by:
max- min
A=
2
If a periodic function has a
maximum value M and a
minimum value m, then the
amplitude of a function is given
by:
max- min
A=
2
Look at the additional
example #1 on page 139.
Stretches and Shrinks:
Vertical stretches and shrinks
y = 2f(x) vertical stretch of 2 times
each y-value
y = ½ f(x) vertical shrink of ½
times each y-value
Stretches and Shrinks:
Vertical stretches and shrinks
y = 2f(x) vertical stretch of 2 times
each y-value
y = ½ f(x) vertical shrink of ½
times each y-value
Therefore, y = c f(x) will
provide a vertical stretch or
vertical shrink of c times each
y-value.
Stretches and Shrinks:
b)Horizontal stretches or shrinks
y = f(2x) horizontal shrink of
½ times each x-value
y = f(½ x) horizontal stretch of
2 times each x-value
Stretches and Shrinks:
b)Horizontal stretches or shrinks
y = f(2x) horizontal shrink of
½ times each x-value
y = f(½ x) horizontal stretch of
2 times each x-value
Therefore, y = f(cx) will provide
a horizontal stretch or shrink of
1/c (reciprocal of c times each
x-values).
These will cause the following changes
to occur in your graph:
These will cause the following changes
to occur in your graph:
If a periodic function f has period p
and amplitude p then:
These will cause the following changes
to occur in your graph:
If a periodic function f has period p
and amplitude p then:
y = c(f(x)) has period p and
amplitude c(A).
These will cause the following changes
to occur in your graph:
If a periodic function f has period p
and amplitude p then:
y = c(f(x)) has period p and
amplitude c(A).
p
y = f(cx) has period
and
c
amplitude A.
Translating graphs
The graphs of y – k = f(x – h) is
obtained by translating the graph of
y = f(x) horizontally
h units and vertically k units.
Translating graphs
The graphs of y – k = f(x – h) is
obtained by translating the graph of
y = f(x) horizontally
h units and vertically k units.
(Take a look at the two
graphs on page 141)
Example:
Example:
Sketch the graph of the following equation a.
Then, using translations, sketch the graphs of b
and c.
Example:
Sketch the graph of the following equation a.
Then, using translations, sketch the graphs of b
and c.
a) y = |x|
b) y – 2 = |x – 3|
c) y = |x + 5|
Look at the chart on page 142.
Use this guidelines as a
reference when working on
homework.
Assignment:
144
all,
Pg. 142C.E. -> 1-6
W.E. -> 1-8 all