Determine amplitude, period, phase shift, and vertical displacement

For sinusoidal functions in the form $y=A\sin(B(x-C))+D$ or $y=A\cos(B(x-C))+D$ , the learner interprets amplitude as $|A|$ , period as $2\pi/|B|$ (or $360^\circ/|B|$ ), phase shift as $C$ , and vertical displacement as $D$ , identifying the midline $y=D$ and the effect of a negative $A$ as reflection. The understanding connects algebraic parameters with graph features and is limited to standard sinusoidal transformations, not advanced Fourier analysis or generalized periodic-function theory.

Detailed Explanation: 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

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: $C$ $C$
- $C>0$ 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},

\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 $A=-3$

Learn by doing: Determine amplitude, period, phase shift, and vertical displacement

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Sinusoidal Function Parameters (4 Params) - Function to Parameters

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