Graphing Trigonometric Functions

Visualizing trigonometric functions through graphs reveals their periodic nature and key characteristics essential for applications in physics, engineering, and signal processing.

Basic Graphs

The six trigonometric functions produce distinct graphical patterns:

Sine (sinθ): "Wave" starting at origin, period 2π
Cosine (cosθ): Wave starting at (0,1), period 2π
Tangent (tanθ): Discontinuous curve with π period

General Form: y = A·sin(B(x - C)) + D
Where:
A = amplitude (vertical stretch)
B = frequency (2π/period)
C = phase shift (horizontal shift)
D = vertical shift

Graphing Example

Graph y = 2sin(3x - π) + 1

Solution:
Amplitude (A) = 2
Period = 2π/3 ≈ 2.094
Phase shift = π/3 to the right
Vertical shift = 1 unit up

Key Features

Feature	Description
Amplitude	Height from midline to peak
Period	Length of one complete cycle
Frequency	Cycles per unit (1/period)
Phase Shift	Horizontal displacement

Transformations

Understanding graph transformations:

Vertical stretch: Changes amplitude
Horizontal compression: Increases frequency
Reflections: Negative coefficients flip graphs

Pro Tip: When graphing tangent functions, first plot the vertical asymptotes at θ = π/2 + πn, then plot points between them.

Real-World Applications

Sound waves (sine graphs represent pure tones)
Tidal patterns (combined sine curves)
AC current voltage over time

Graphing transforms abstract equations into visual patterns, enabling intuitive understanding of trigonometric behavior.

📈 Graphing Trigonometric Functions

Basic Graphs

Graphing Example

Key Features

Transformations

Real-World Applications