Graphing Trigonometric Functions — Mathematics Knowledge System

Overview

The graphs of sine, cosine, and tangent are periodic waveforms whose shape is controlled by four parameters in the general form y = A sin(Bx + C) + D: the amplitude A (half the vertical distance between max and min), the period 2π/|B| (horizontal length of one full cycle), the phase shift −C/B (horizontal translation), and the vertical shift D (midline height). Tangent, cotangent, secant, and cosecant have their own characteristic shapes, including vertical asymptotes where the underlying function is undefined. Understanding how each parameter transforms the base graph is essential for modeling periodic phenomena like tides, sound waves, and seasonal temperature cycles.

Intuition

Think of the base graph y = sinx as a wave template, and each parameter as a specific transformation applied to it: A stretches or compresses it vertically, B stretches or compresses it horizontally (a larger B squeezes the wave into a shorter period), C slides it left or right, and D lifts or lowers the whole thing. Because sine and cosine are the same curve shifted by 90° (cosx = sin(x+π/2)), any cosine graph can be redrawn as a phase-shifted sine graph and vice versa — the 'phase shift' parameter is exactly what lets you convert between them.

Interactive Graph

Formal Definition

Definition

For the general sinusoidal function:

y = A\sin(Bx + C) + D

General sine form

\text{Amplitude} = |A|, \quad \text{Period} = \frac{2\pi}{|B|}, \quad \text{Phase shift} = -\frac{C}{B}, \quad \text{Midline: } y = D

Parameters

y = \tan x \text{ has vertical asymptotes at } x = \frac{\pi}{2} + n\pi,\ n\in\mathbb{Z}

Tangent asymptotes

Worked Examples

Match to y = Asin(Bx+C)+D: A=2, B=3, C=−π/2, D=1.
$A=2,\ B=3,\ C=-\frac{\pi}{2},\ D=1$
Compute period and phase shift.
$\text{Period} = \frac{2\pi}{3}, \quad \text{Phase shift} = -\frac{-\pi/2}{3} = \frac{\pi}{6}$

Answer: Amplitude 2, period 2π/3, phase shift π/6 (right), midline y=1

Practice Problems

Difficulty 3/10

Find the amplitude, period, and midline of y = −3cos(2x) + 4.

Difficulty 4/10

Where are the vertical asymptotes of y = tan(2x) in the interval (0, π)?

Difficulty 6/10

Tide height (m) is modeled by h(t) = 3sin(π t/6) + 5, t in hours. Find the amplitude, period, and the maximum tide height.

Quiz

Summary

y = Asin(Bx+C)+D has amplitude |A|, period 2π/|B|, phase shift −C/B, and midline y=D.
Tangent, cotangent, secant, and cosecant graphs have vertical asymptotes where the underlying sine or cosine is zero.
Real periodic phenomena (tides, AC voltage, sound waves) are modeled by fitting these four parameters to data.

References

WebsiteWikipedia — Trigonometric functions