6-2 Amplitude, Period, and Phase Shift
By Faith (Honors ICM)
Objectives: After reviewing this lesson, you should be able to:
find the amplitude, period, and phase shift for a trigonometric function, and
write equations of trigonometric functions given the amplitude, period, and phase shift
Definitions:
| Amplitude: | |A| for functions in form y = A sin ka or y = A cos ka |
* The amplitude can also be described as the absolute value of one- half the difference of the maximum and minimum functions values
|A- (-A)| = |A|
|2|
* The tangent, cotangent, secant, and cosecant functions do not have amplitudes because their values increase and decrease without bound for values that approach certain multiples of 90°.
| Period: | the least positive value of a for which f(X) = f(x + a) |
*Therefore the period of y = sin kØ is 360°/k. Similary, the period of y = cos kØ is 360°/k. The period of y = tan kØ is 180°/k.
| Phase Shift: | the least value of |kØ + c|, for which the trigonometric function f(kØ+c)= 0 |
*The phase shift of the function y= sin (kØ + c) is -c/k. If c> 0, the shift is to the left. If c< 0, the shift is to the right, this definition applies to all of the trigonometric functions.
Examples:
State the amplitude, period, and phase shift for each function.
1) y = -7sin 8Ø
| Amplitude= | |A| » |-7| = 7 (7, is how high and low the curve will ever go on this particular graph) |
| Period= | sine function period, y = sin kØ is 360º/k.» 360°/8 = 45° or pi/4 |
| Phase Shift= | *there is no c for a phase shift |
2) y= tan (6Ø - 90°)
| Amplitude= | tangent functions do not have amplitudes because their values increase and decrease without bound |
| Period= | tan function period, y = tan kØ is 180°/k.» 180°/6 = 30º or pi/6 |
| Phase Shift= | y = A sin (kØ + c) is -c/k, -(-90º/6) = 15º |
Write an equation of each function.
3) a sine function with amplitude 5, period 360°, and phase shift 30°
| y = ±5sin (Ø - 30°) | Phase Shift= 30*1 (Ø), instead of -c/k to find shift
Ø = 360° (same period as the function) |
4) a tangent function with period of 720° and phase shift of 60°
| y = ±tan (¼Ø - 15º) | Period= 720º/360º= ¼
Phase Shift= 60*¼= 15º, instead of -c/k |
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