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Chapter four
AMPLITUDE, PERIOD,
AND PHASE SHIFT
Introduction
In this chapter, you and your students will learn how to modify trigonometric
functions by constants. If you multiply sine or cosine by a constant A, then
every value of the function is multiplied by A. The maximum absolute value of
the sine or cosine function is 1. Therefore, Asin
maximum value of |A|. The value of |A| is called the amplitude of the functions
y = Asin x and y = Acos x, and is the distance sine and cosine rises and falls
above the x-axis. Multiplication of the function by A will stretch or shrink the
curve vertically. |A| > 1 will stretch the curve while an |A| between 0 and 1
will shrink the curve. The tangent function does not have an amplitude because
tangent increases without bound as
If we multiply the argument
then every measure of an angle
stretch or shrink the curve horizontally. A value of k > 1 will shrink the curve
while a value of k between 0 and 1 will stretch the curve. The stretching and
shrinking of a curve horizontally effects the function's period. The new periods
2
will be
for sine, cosine, secant and cosecant, and
k
14
Amplitude, Period, and Phase Shift/TRIGONOMETRY USING THE SHARP EL-9600
approaches values such as
by a constant k within a trigonometric function,
is multiplied by k. Multiplication of
and Acos
will have a
.
2
for tangent and cotangent.
k
will
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