Graphs of Trigonometric Functions

Amplitude, Period, and Phase Shift

In this lesson:

• Identify base graphs of sin, cos, and tan
• Identify amplitude, period, phase shift, and vertical shift
• Match graphs and equations in both directions

What You Will Learn Today

1. Match a graph with its equation: midline, amplitude, period, phase shift
2. Identify base graphs of sin, cos, tan: key points, period, range
3. Identify , , , in sine or cosine form; tan period
4. Identify a transformed graph from its equation

Unit Circle Values Become the Wave

In radians, an angle larger lands on the same point, so values repeat every .

The Base Graph of Sine

Zero, max, zero, min, zero. Amplitude , period , range .

The Base Graph of Cosine

Max, zero, min, zero, max. Range . : sine shifted left .

The Base Graph of Tangent

Period , range all reals, asymptotes at , no amplitude.

General Form of a Sinusoidal Function

Amplitude Stretches the Wave Vertically

• reaches and
• reaches and
• flips over the midline

: amplitude , not .

Period Controls the Horizontal Width

Period for sin and cos

• : period , two cycles in
• : period , half a cycle in
• : period

Larger compresses; smaller stretches.

Tangent Period Is Pi Over B

: period , not . : no amplitude, period .

Phase Shift Requires Dividing by B

shifts right . In , is the shift.

For : shift , not just .

Example: . Shift : left , not .

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
• Shift (right)
• : midline
• 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
• Check: at ,

Example: Sine Form With Phase Shift

Max , min , peaks apart, upward crossing at .

• , ,
• Shift right,

Check: at , .

Equation to Graph: Predict the Features

Match a graph on midline, range, period, then position.

ACT Strategy: Eliminate by Amplitude First

Graph: max , min , peak at , period .

• (a)
• (b)
• (c)
• (d)

Amplitude eliminates (a), (d); the peak at eliminates (b). Radian mode: .

Key Takeaways and Common Mistakes

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

Watch out:

• Tangent has no amplitude
• Peak to trough is half the period
• Negative reflects, does not shift
• Radian mode on the calculator

Coming Up Next in Trigonometry

Up next: Pythagorean identity and basic trig identities

• Other basic identities
• Simplifying trig expressions