Graph of sin²(3x) Explained

Grades 11-12

Question: 〖sin^2 (3x), (0≤x≤⁡2x)

The function

f(x) = \sin^2(3x)

is a trigonometric function that incorporates horizontal compression and a vertical transformation (squaring). To understand the behavior of this graph on the interval

[0, 2\pi]

, we can break down its components.

1. Horizontal Compression

The coefficient

3

inside the sine function,

\sin(3x)

, increases the frequency of the oscillation. While a standard

\sin(x)

curve has a period of

2\pi

, the function

\sin(3x)

completes one full cycle every

\frac{2\pi}{3} \approx 2.09

units.

2. The Effect of Squaring

Squaring the function has two major effects:

Range: Since any real number squared is non-negative, the range of the function shifts from $[-1, 1]$ to $[0, 1]$ . The graph never dips below the x-axis.
Periodicity: Squaring the sine wave effectively doubles the frequency of the peaks because both the original positive and negative peaks of $\sin(3x)$ become positive maximums at $y=1$ .

We can see this mathematically using the double-angle identity for power reduction:

\begin{align*} \sin^2(\theta) &= \frac{1 - \cos(2\theta)}{2} \\ \sin^2(3x) &= \frac{1 - \cos(6x)}{2} \\ \text{The period } T \text{ is given by: } T &= \frac{2\pi}{6} = \frac{\pi}{3} \end{align*}

3. Key Features of the Graph

Period: The graph repeats every $\frac{\pi}{3} \approx 1.05$ units. Over the interval $[0, 2\pi]$ , you will see exactly 6 complete cycles.
Maximum Points: The peaks reach a height of $y = 1$ . These occur at $x = \frac{\pi}{6}, \frac{\pi}{2}, \frac{5\pi}{6}, \dots$
Minimum Points (Roots): The troughs touch the x-axis ( $y = 0$ ) at $x = 0, \frac{\pi}{3}, \frac{2\pi}{3}, \dots$

In the interactive diagram below:

The blue curve represents $f(x) = \sin^2(3x)$ .
The orange segment on the x-axis highlights one single period of the function ( $\pi/3$ ).
The dotted gray line indicates the maximum value of $1$ .
You can move the slider to trace the red point $P$ along the curve and observe its coordinates.

法提供 GeoGebra 代码。