Grades Algebra II & Trigonometry Trigonometric FunctionsGraphs of Sine, Cosine & Tangent

Graphs of Sine, Cosine & Tangent

🌊 Trigonometric Functions

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Graphs of Sine, Cosine & Tangent

Analyze amplitude, period, phase shift, and vertical shift of trig functions.

The general sinusoidal function y = A sin(B(x − C)) + D has amplitude |A|, period 2π/|B|, phase shift C, and vertical shift D. Understanding these parameters lets you graph any sinusoidal function quickly.

y = A\sin(B(x - C)) + D \qquad \text{Amplitude} = |A|,\quad \text{Period} = \frac{2\pi}{|B|}

Key Properties

Amplitude |A|: distance from midline to maximum (or minimum)Period 2π/|B|: length of one complete cyclePhase shift C: horizontal shift (right if C > 0)Vertical shift D: the midline y = D

y = \tan(x): \text{ period } \pi,\text{ vertical asymptotes at } x = \frac{\pi}{2} + n\pi,\; n \in \mathbb{Z}

📈Analyzing y = 3sin(2x − π/4) + 1

A = 3 → amplitude 3. B = 2 → period = 2π/2 = π. Phase shift = π/4 (right). D = 1 → midline y = 1. Maximum value: 1 + 3 = 4. Minimum: 1 − 3 = −2.

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Remember This!

For cosine, y = A cos(B(x − C)) + D. Cosine starts at the maximum when x = C (where the phase shift brings you to x = 0 of the base function). Sine starts at the midline.

✏️ Try It!

What is the period of y = sin(4x)?

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