Graphing Trig Functions and Their Transformations
Graph all six trig functions with full transformations: A·f(Bx − C) + D. Identify amplitude (|A|), period (2π/|B| for sine/cosine), phase shift (C/B), and vertical shift (D) from equations and write equations from graphs.
A lesson of Pre-Calculus, in Trigonometry: Angles and Functions, about 10 minutes.
By the end of this lesson, you will be able to graph any transformation of $\sin$, $\cos$, $\tan$, $\sec$, $\csc$, and $\cot$ with correct amplitude, period, phase shift, and vertical shift.
Complete Summary of Trig Graph Parameters
For $y = a\sin(b(x-h)) + k$ or $y = a\cos(b(x-h)) + k$:
| Parameter | Meaning |
|---|---|
| $a$ | Amplitude $= |
| $b$ | Period $= 2\pi/ |
| $h$ | Phase shift right by $h$ |
| $k$ | Vertical shift; midline $y = k$ |
For $y = a\tan(b(x-h)) + k$: period $= \pi/|b|$.
Graphing Strategy: Key Points
For $y = a\sin(bx)$: divide one period into four equal parts and track:
- Zero → max → zero → min → zero (sin)
- Max → zero → min → zero → max (cos)
Then shift using $h$ and $k$.
Example 1 — Complex Sinusoidal
$y = -3\cos\left(\frac{\pi}{2}x + \frac{\pi}{4}\right) + 1$
Write as $-3\cos\left(\frac{\pi}{2}\left(x + \frac{1}{2}\right)\right) + 1$:
$a = -3$ (amplitude 3, reflected); $b = \pi/2$ (period $= 2\pi/(\pi/2) = 4$)
Phase shift: $h = -1/2$ (left $1/2$); Vertical shift: $k = 1$
Starting point: $x = -1/2$ (where argument = 0)
Key points:
| $x$ | Argument | $\cos$ | $-3\cos + 1$ |
|---|---|---|---|
| $-1/2$ | $0$ | $1$ | $-2$ |
| $1/2$ | $\pi/2$ | $0$ | $1$ |
| $3/2$ | $\pi$ | $-1$ | $4$ |
| $5/2$ | $3\pi/2$ | $0$ | $1$ |
| $7/2$ | $2\pi$ | $1$ | $-2$ |
Midline: $y = 1$; Max: $4$; Min: $-2$.
Example 2 — Tangent with Transformation
$y = 2\tan(3x - \pi) - 1$