“**A****”
controls the height or the amplitude of the curve as shown in the 3 examples
below:
**

*Practice:*

y = 0.9 sin(x) will stretch/shrink the height of the sine graph by a factor of __________.

The amplitude of this graph is:________________.

y = ¼ sin(x) will stretch/shrink the height of the sine graph by a factor of __________.

The amplitude of this graph is:________________.

y = 5 sin(x) will stretch/shrink the height of the sine graph by a factor of __________.

The amplitude of this graph is:________________.

===========================================================

“**B**”
controls the length of one cycle, or the period of the curve. The
period of the sine curve y = sin(x) is _{
}. The period of the sine curve **y = sin Bx**
is _{
}. This is because the number B stretches or shrinks the curve
in the x direction. If the number B is bigger than 1, the period
will shrink. If the number B is smaller than 1, the period will stretch.
Notice that B also gives the number of cycles in the interval from [0, _{
}].

*Practice:*

The length of one period in y = sin(x) is__________________________
since there is/are _________ cycles in the interval from [0, _{
}].

.

The length of one
period in y = sin (3x) is ________________________ since there is/are _________
cycles in the interval from [0, _{
}].

.

The length of one
period in y = sin [(1/2)x] is______________________ since there is/are
_________ cycles in the interval from [0, _{
}].

.

==============================================================

“**C**”
controls the phase shift, or the amount the graph shifts to the left or
to the right. The phase shift is calculated by:

_{ }
shifts _{ }
units to the ** left**.

_{ }
shifts _{ }
units to the *right.*

It looks the same as the prior graph because it is the same!

_{ }
= _{ }
=

What if shifts and stretches or shrinks occur
simultaneously? Example: _{ }
= _{ }
period shrinks by a factor of 5. This means that the new period is _{
}. The graph shifts _{ }
units to the *left.*

*Practice:*

The phase shift in the curve y = sin(3x-6) is ____. It shifts the curve ____ units to the _________. Also the period will shrink/stretch by a factor of ____. The new period is__________.

The phase shift in the curve y = sin( 0.5x + _{
}) is__________. It shifts the curve ______________ units to
the ___________. Also the period will shrink/stretch by a factor
of**___________.** The new period is__________.

The phase shift in the curve y = sin( 6x - 3_{
}) is_________. It shifts the curve __________ units to the
____________** .** Also the period will shrink/stretch by
a factor of _____________. The new period is_______________..

========================================================

**Answers to Practice:**

**A****.
is for ****A**mplitude.

y = 0.9 sin(x) will stretch/shrink the height
of the sine graph by a factor of __0.9__.

The amplitude of this graph is: __0.9__

y = ¼ sin(x) will stretch/shrink the height
of the sine graph by a factor of __¼__

The amplitude of this graph is: __
¼__.

y = 5 sin(x) will stretch/shrink the height of
the sine graph by a factor of __5__ .

The amplitude of this graph is: __5__

**B**.
is for **B**ulge (stretch or shrink the size of the Period)!

The length of one period in y = sin(x) is _{ }
since there is **1 cycle** in the interval from [0, _{
}].

The length of one period in y = sin (3x) is _{ }
since there are __3 cycles__ in the interval from [0, _{
}].

The length of one period in y = sin [(1/2)x] is _{ } __
__since there is __½ of a cycle__ in the interval from [0, _{
}].

**C****.
is for ****C**hange (shift) in location:

The phase shift in the curve y = sin(3x-6) is
2. It shifts the curve 2 units to the right. Also the period
will shrink/stretch by a factor of 3. The new period is _{
}.

The phase shift in the curve y = sin( 0.5x + _{
}) is _{
}. It shifts the curve _{ }
units to the left (which means it doesn’t go anywhere!) . Also the
period will shrink/** stretch** by a factor of

The phase shift in the curve y = sin( 6x - 3_{
}) is _{
}. It shifts the curve _{ }
units to the ** right.** Also the period will