You need to know what
the three terms amplitude, period, and
phase shift mean,
in the context of talking about the generalized sine and cosine curves shown below:
Generalized Sine/Cosine Curves |
As discussed in the prior section, Graphing Generalized Sines and Cosines, each of these curves is the result of starting with either $\,y = \sin x\,$ or $\,y = \cos x\,$, and then applying graphical transformation(s) chosen from: These transformations affect the ‘height’ (amplitude), ’width’ (period), and ‘left/right positioning’ (phase shift) of the resulting curve. The ideas are made precise below. |
|
$\color{red}{y = a\sin k(x + b)}$ | $\color{red}{y = a\cos k(x + b)}$ | |
$\color{red}{y = a\sin (kx + B)}$ | $\color{red}{y = a\cos (kx + B)}$ | |
In these equations:
|
As discussed below,
given any generalized sine or cosine curve,
you should be able to
determine its amplitude, period, and phase shift.
Sample question:
State the amplitude, period, and phase shift of $\,y = 5\sin(3x-1)\,$.
In the next section, you will write an
equation of a curve with a specified amplitude, period, and phase shift.
Sample question:
Write an equation of a sine curve with amplitude $\,5\,$, period $\,3\,$, and phase shift $\,2\,$.
AMPLITUDE:
|
![]() |
PERIOD:
|
![]() ![]() |
PHASE SHIFT:
|
![]() |
The argument is of the form $\,kx + B\,$.
The expression is of the form $\,a\sin(kx+B)\,$. Here, $\,a = 5\,$, $\,k = 3\,$ and $\,B = -1\,$. The amplitude is $\,|a| = |5| = 5\,$. The period is $\,\displaystyle\frac{2\pi}{|k|} = \frac{2\pi}{3}\,$. Setting $\,3x-1 = 0\,$ gives $\,\displaystyle x = \frac{1}{3}\,$; the phase shift is $\,\displaystyle\frac{1}{3}\,$. Alternatively, the phase shift is: $\,\displaystyle\frac{-B}{k} = \frac{-(-1)}3 = \frac{1}{3}\,$ |
![]() |
The argument is of the form $\,k(x + b)\,$.
The expression is of the form $\,a\cos k(x + b)\,$. Here, $\,a = -3\,$, $\,k = 2\,$ and $\displaystyle\,b = \frac{\pi}{5}\,$. The amplitude is $\,|a| = |-3| = 3\,$. The period is $\,\displaystyle\frac{2\pi}{|k|} = \frac{2\pi}{2} = \pi\,$. Setting $\displaystyle\,2(x+\frac{\pi}{5}) = 0\,$ gives $\,\displaystyle x = -\frac{\pi}{5}\,$; the phase shift is $\,\displaystyle -\frac{\pi}{5}\,$. Alternatively, the phase shift is: $\displaystyle\,-b = -\frac{\pi}{5}\,$ |
![]() |
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. |
PROBLEM TYPES:
|