Amplitude, Period, and Phase Shift- Trigonometry Guide

What You're Actually Looking At

When you see a sine or cosine wave, you're looking at three things that determine its shape and position. Amplitude tells you how tall it is. Period tells you how wide one complete wave is. Phase shift tells you where it starts horizontally.

Most textbooks make this sound complicated. It's not. Once you see the numbers in the equation, you can predict exactly what the graph will do.

Amplitude: The Height

Amplitude is the distance from the midline of the wave to its maximum or minimum point. It's always positive, even if the graph goes below the x-axis.

In the equation y = A sin(x) or y = A cos(x), the amplitude is the absolute value of A.

Examples:

• y = 3 sin(x) has amplitude 3
• y = -2 cos(x) has amplitude 2 (the negative flips the graph, but amplitude stays positive)
• y = ½ sin(x) has amplitude 0.5

The midline sits at y = 0 unless there's a vertical shift. We'll get to that.

Period: The Width of One Wave

The period is the horizontal distance needed to complete one full cycle of the wave.

For basic sine and cosine, the period is 2π. When you multiply x by a number, you compress or stretch the wave horizontally.

The formula is simple:

Period = 2π / B (when the equation is y = sin(Bx) or y = cos(Bx))

Examples:

• y = sin(2x) has period 2π/2 = π
• y = cos(4x) has period 2π/4 = π/2
• y = sin(x/3) has period 2π / (1/3) = 6π

When B is greater than 1, the wave compresses horizontally. When B is between 0 and 1, it stretches out.

Phase Shift: Where It Starts

Phase shift moves the graph left or right. It happens when you add or subtract a value inside the parentheses with x.

The formula:

Phase Shift = -C / B (when the equation is y = sin(Bx - C) or y = cos(Bx - C))

Examples:

• y = sin(x - π/2): phase shift = π/2 to the right
• y = cos(x + π): phase shift = π to the left (because +π inside means subtracting π)
• y = sin(2x - π): phase shift = π/2 to the right (because -π divided by 2 = -π/2, then negative of that = π/2)

Watch the sign carefully. The formula subtracts C, so a positive C inside the parentheses shifts right, and a negative C shifts left.

Putting It All Together

The complete trig function looks like this:

y = A sin(Bx - C) + D

Each letter controls something specific:

• A = amplitude (vertical stretch)
• B = period (horizontal stretch/compression)
• C = phase shift (horizontal movement)
• D = vertical shift (moves the midline up or down)

Vertical Shift

D moves the entire graph up or down. The midline becomes y = D instead of y = 0. This doesn't affect amplitude or period, just where the wave sits vertically.

Quick Comparison Table

How to Find All Three: Step by Step

Given: y = 3 sin(2x - π) + 1

Step 1: Amplitude

Take the absolute value of A. Amplitude = |3| = 3.

Step 2: Period

Use 2π divided by B. Period = 2π/2 = π.

Step 3: Phase Shift

Rewrite the inside as B(x - C/B). Here: 2x - π = 2(x - π/2). Phase shift = π/2 to the right.

Step 4: Vertical Shift

D = 1. The midline is at y = 1.

That's it. Four numbers, four properties.

Common Mistakes to Avoid

• Forgetting to divide C by B when calculating phase shift. The shift is always relative to the coefficient of x.
• Thinking a negative amplitude means negative amplitude. It doesn't. It means the graph is reflected.
• Confusing phase shift with period. Phase shift moves the graph. Period changes how squished or stretched it is.
• Ignoring the vertical shift when finding the range. The range is [D - |A|, D + |A|], not [-A, A].

Getting Started: Identifying from a Graph

Look at the highest and lowest points. The distance from the midline to either extreme is the amplitude. Count the x-axis distance for one full wave to get the period. See where a peak or trough lines up with the y-axis to estimate phase shift.

Practice with a few graphs and you'll start reading these properties automatically. The equations just tell you what the graph already shows.