Graphs of Trigonometric Functions

Amplitude, Period, and Phase Shift

In this lesson:

• Sketch base graphs of sin, cos, and tan
• Identify amplitude, period, and phase shift
• Write equations to match given graphs

What You Will Learn Today

After this lesson, you will:

1. Sketch base graphs of sin, cos, tan
2. Identify amplitude, period, phase shift, vertical shift
3. Graph a transformed sinusoidal function
4. Read a graph to find key features
5. Write an equation matching a given graph

Why Do Trig Functions Make Waves?

On the unit circle, is the y-coordinate as sweeps around.

• At : height is
• At : height is
• At : height is again
• At : height is

Plot those heights against — you get a wave.

The Base Graph of Sine

Five key points in : zero, max, zero, min, zero.

The Base Graph of Cosine

Five key points: max, zero, min, zero, max. Same shape, different start.

The Base Graph of Tangent

Period is , not . Tangent has no amplitude.

Comparing the Three Base Graphs

Quick Check: Identify the Function

A trig graph starts at its maximum value and oscillates between and with period .

Which function is it?

Think about it before advancing...

General Form of a Sinusoidal Function

Amplitude Stretches the Wave Vertically

Compare , , and :

• : wave reaches and
• : wave reaches and
• : wave flips over the midline

Amplitude , always positive.

Period Controls the Horizontal Width

Period for sin and cos; for tan

• : period (two cycles in )
• : period (half a cycle in )

Larger compresses; smaller stretches.

Phase Shift Requires Dividing by B

Phase shift (not just )

Example:

Factor out :

Phase shift (left by )

Not . Always divide by first.

Vertical Shift Moves the Midline

shifts the midline to .

• Range becomes
• Example: oscillates between and

The midline is not the x-axis anymore.

Worked Example: All Four Parameters Combined

• → amplitude
• → period
• Phase shift (right)
• → midline at
• Range:

Quick Check: Find All Four Parameters

Find the amplitude, period, phase shift, and midline.

Work it out before advancing...

Four Steps From Graph to Equation

Example: Reading a Graph for Cosine

Max , min , peaks at .

• , , period ,
• Peak at → cosine, no shift

Example: Sine Form With Phase Shift

Max , min , upward crossing at .

• ,
• Period , so
• Shift right,

ACT Strategy: Eliminate by Amplitude First

Which equation matches the graph?

• A)
• B)
• C)
• D)

Check amplitude → eliminates A, D. Then period → confirms .

Practice: Find Parameters From These Graphs

Problem 1: Max , min , period . Find , , .

Problem 2: . Find amplitude, period, shift.

Problem 3: Peaks , troughs , period , peak at . Write the equation.

Practice Answers and Solutions Revealed

Problem 1: , ,

Problem 2: Amplitude , period , phase shift right

Problem 3: , ,

Key Takeaways and Common Mistakes

• Period (sin/cos), (tan)
• Phase shift — always divide by

Watch out:

• Tangent has no amplitude
• Peak to trough period, not full
• Negative reflects, does not shift

Coming Up Next in Trigonometry

Up next: Trigonometric identities and equations

• Pythagorean identities:
• Solving trig equations algebraically
• Connecting identities to graph features