Page 309 – Amplitude, Period and Phase Shift

Objective

•To find the amplitude, period and phase shift for a trigonometric function •To write equations of trigonometric functions given the amplitude, period, and phase shift

Glossary

• Amplitude • Period

• Phase Shift

Amplitude of Sine and Cosine Functions

The amplitude of the functions y = A sin and y = A cos is the absolute value of A

The tangent, cotangent, secant and cosecant functions do not have amplitudes because their values increase and decrease without bound.

State the amplitude of the function y = 3 cos Graphy 3 cos y = cos

on the same set of axes. Compare the graphs.

According to the definition of amplitude, A = 3.

Make a table of values.

0° 45° 90° 135°

180°

225° 270°

315°

360°

cosQ 1 .71 0 -.71 -1 -.71 0 .71 13cosQ 3 2.1

20 -

2.12-3 -2.12 0 2.1

23

Graph the points and draw a smooth curve.

Period

• The period of a function is the distance on the x-axis it takes a function to go through one complete cycle.

The period of the functions y = sin k and y = cos k

is: 360° k

The period of the function y = tan k is: 180° k

State the period of the function y = sin 4. Then graph the function and

y = sin on the same set of axes.

By definition, the period of the sin function is 360°/k.

Period = 360°/4 = 90°

This means the function y = sin 4 goes through one complete cycle in 90°.

Phase Shift

• Phase shift moves the graph of the function horizontally.

The phase shift of the function y = A sin (k+ c) is: - c

k

If c > 0 the shift is to the left. If c < 0 the shift is to the right.

This applies to all the trigonometric functions.

State the phase shift of the function y = tan ( – 45). Then graph the function

and y = tan on the same axes and compare.

The phase shift is – c/k.

- - 45 1

= 45°

Since c is less than 0 the shift is to the right.

Assignment

• Page 315– # 4 – 11, 15 - 24