Lesson 4-4Lesson 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:

If a periodic function has a maximum value M and a

minimum value m, then the amplitude of a function is given

by:

Look at the additional example #1 on page 139.

Stretches and Shrinks:Vertical stretches and shrinksy = 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 shrinksy = 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).

y = f(cx) has period and

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:

Pg. 142-144  C.E. -> 1-6 all,

W.E. -> 1-8 all