Lesson 4.6

Phase Shifts

A phase shift slides the entire wave left or right along the -axis. It changes where the cycle starts without changing shape, height, or speed.

Introduction

Not every oscillation starts at the same moment. A delayed echo, a shifted electrical signal — all modeled by a horizontal shift of a sine or cosine wave.

Past Knowledge

You can graph waves with custom amplitude, period, and vertical shift.

Today's Goal

Identify and apply phase shifts in .

Future Success

Phase shift is the last piece of .

Key Concepts

How Phase Shifts Work

in : Shift right
(i.e. ): Shift left

Subtract = right, Add = left.

Gray = sin(x) · Blue = sin(x − π/4) · Red = sin(x + π/3)

When is Present

If the equation is , factor out first:

Phase shift

Worked Examples

Basic

Identifying the Phase Shift

Question: What is the phase shift of ?

Step 1: → shift right.

Final Answer: Right units.

Intermediate

Factoring Out B

Question: Find the period and phase shift of .

Step 1: Factor:

Step 2: Period . Phase shift right.

Final Answer: Period , shift right .

Advanced

Five Key Points with Phase Shift

Question: List the five key points of .

Step 1: Period . Cycle starts at .

Step 2: Quarter-period . The -values:

Step 3: Pattern:

Final Answer: Sine pattern shifted right by .

Common Pitfalls

Direction Backwards

shifts right, not left. “Subtract = later = right.”

Forgetting to Factor B

In , the shift is , NOT . Always factor first.

Real-Life Applications

Daylight Hours

Daylight hours follow a sinusoidal pattern, but the peak isn't January 1 — it's around June 21. This offset from the calendar start is modeled by a phase shift: .