Lesson 94: Graphing Trigonometric Functions
Graphing sine, cosine, and tangent functions helps visualize periodic behavior, amplitude, period, and phase shift. Understanding these graphs is essential for modeling waves, oscillations, and cycles in real-world contexts.
Sine and Cosine Functions
General form:
- y = a sin(bx + c) + d
- y = a cos(bx + c) + d
- Amplitude = |a| (height from centerline to peak)
- Period = 2Ï€ / b (length of one full cycle)
- Phase shift = −c / b (horizontal shift)
- Vertical shift = d (centerline of wave)
Example: y = 2 sin(x − π/4) + 1 → amplitude = 2, period = 2π, phase shift = π/4 right, vertical shift = 1.
Tangent Function
General form: y = a tan(bx + c) + d
- Period = π / b
- Vertical asymptotes occur where cos(bx + c) = 0
- Amplitude is not defined (tangent extends to infinity)
- Phase and vertical shifts are handled as in sine/cosine
Example: y = tan(x − π/6) → vertical asymptotes at x = π/6 + nπ, n ∈ ℤ.
Examples
Example 1 — Sine Graph: Graph y = 3 sin(2x) → amplitude = 3, period = π, no phase or vertical shift.
Example 2 — Cosine Graph: Graph y = −2 cos(x − π/3) + 1 → amplitude = 2, period = 2π, phase shift = π/3 right, vertical shift = 1, reflection over x-axis.
Example 3 — Tangent Graph: Graph y = tan(x + π/4) → vertical asymptotes at x = −π/4 + nπ, n ∈ ℤ, period = π.
Practice Problems
- Graph y = sin(x) and label one full cycle.
- Graph y = 2 cos(x − π/2) and identify amplitude, period, phase, and vertical shift.
- Graph y = −tan(3x) and identify vertical asymptotes and period.
- Graph y = 0.5 sin(2x + π/3) − 1 and describe transformations from y = sin(x).
- Find the period and phase shift of y = cos(4x − π/2).