Skill: Determine amplitude, period, phase shift, and vertical displacement

To identify the features of a sinusoidal function, compare it with one of these forms:

$$y=A\sin(B(x-C))+D$$

or

$$y=A\cos(B(x-C))+D.$$

The parameters have these meanings:

• Amplitude: $\lvert A\rvert$
• Period: $\dfrac{2\pi}{\lvert B\rvert}$ radians, or $\dfrac{360^\circ}{\lvert B\rvert}$ degrees
• Phase shift: $C$
  • $C>0$ means a shift right.
  • $C<0$ means a shift left.
• Vertical displacement: $D$
• Midline: $y=D$
• If $A<0$, the graph is reflected across its midline.

Example

Determine the amplitude, period, phase shift, and vertical displacement of

$$y=-3\cos\left(2\left(x-\frac{\pi}{4}\right)\right)+2.$$

Step 1: Identify the parameters

Compare the function with

$$y=A\cos(B(x-C))+D.$$

We can see that

$$A=-3,\qquad B=2,\qquad C=\frac{\pi}{4},\qquad D=2.$$

Step 2: Find the amplitude

The amplitude is the absolute value of $A$:

$$\text{Amplitude}=\lvert -3\rvert=3.$$

The negative sign does not make the amplitude negative. Instead, it reflects the cosine graph across its midline.

Step 3: Find the period

Using radians,

$$\text{Period}=\frac{2\pi}{\lvert B\rvert} =\frac{2\pi}{2} =\pi.$$

So the graph completes one full cycle every $\pi$ units.

Step 4: Find the phase shift

The expression is

$$x-\frac{\pi}{4},$$

so

$$\text{Phase shift}=\frac{\pi}{4}.$$

The graph shifts $\frac{\pi}{4}$ units to the right.

Step 5: Find the vertical displacement and midline

The value of $D$ is $2$, so

$$\text{Vertical displacement}=2.$$

The midline is therefore

$$y=2.$$

Thus, the function has:

• Amplitude: $3$
• Period: $\pi$
• Phase shift: $\frac{\pi}{4}$ units right
• Vertical displacement: $2$ units up
• Midline: $y=2$
• Reflection: Yes, because $A=-3$