Amplitude, Period, and Phase Shift

One Formula, Four Knobs

The plain wave is $y = \sin(x)$. But real waves (sound, light, tides) are taller, faster, or shifted over.

We control all of that with four numbers:

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

Each letter is a knob you can turn. Let’s take them one at a time.

Amplitude: How Tall (A)

A stretches the wave vertically. It sets how far the wave reaches above and below its midline.

• $A = 1$: normal wave, peaks at 1
• $A = 3$: peaks at 3, three times as tall
• $A = 0.5$: shallow wave, peaks at 0.5

The amplitude is the distance from the midline to a peak. It equals $|A|$.

A bigger $A$ means a louder sound or a taller ocean swell.

Period: How Wide (B)

B squeezes or stretches the wave horizontally. It controls how quickly the wave repeats.

The bigger $B$ is, the more cycles get packed into the same space.

The period is the width of one full cycle: $\text{period} = \frac{2\pi}{B}$

Double $B$ and you halve the period: the wave repeats twice as fast.

Phase Shift: Slide Left or Right (C)

C slides the whole wave sideways. Nothing about its shape changes, it just starts somewhere else.

• A positive $C$ shifts the wave to the right
• A negative $C$ shifts it to the left

The phase shift is $C$: where the cycle begins.

This is exactly the trick from the last lesson. Cosine is sine with a phase shift.

Vertical Shift: Raise or Lower (D)

D moves the entire wave up or down. It sets the midline, the horizontal line the wave is centered on.

• $D = 0$: waves around the x-axis
• $D = 5$: waves around the line $y = 5$

The wave keeps its shape. Only its center moves.

Putting It Together

To read any wave, match it to $y = A\sin(B(x - C)) + D$.

Example

$y = 3\sin\big(2(x - \tfrac{\pi}{4})\big) + 1$

Match the knobs:

• $A = 3$: peaks reach 3 above the midline
• $B = 2$: period is $\frac{2\pi}{2} = \pi$
• $C = \frac{\pi}{4}$: shifted right by $\frac{\pi}{4}$
• $D = 1$: midline is $y = 1$

So this wave is centered on $y = 1$, reaches from -2 up to 4, completes a cycle every $\pi$, and starts that cycle at $x = \frac{\pi}{4}$.