Graph of a Periodic Function with Period π/2 (y = sin 4x)

Grades 10-11

Question: Graph the periodic function with period 2*pi/4.

To graph a periodic function with period

T = \frac{2\pi}{4}

, we first need to understand the relationship between the period and the angular frequency of a basic trig function like sine or cosine.

The general form of a sine function is

y = \sin(Bx)

, where the period

T

is given by:

\begin{align*} T = \frac{2\pi}{B} \end{align*}

Here, we're asked for a period of

T = \frac{2\pi}{4}

. Comparing the two expressions, it's clear that

B

must equal

4

. So the function we'll graph is:

\begin{align*} f(x) = \sin(4x) \end{align*}

Analyzing the graph:

Horizontal compression: Unlike the standard $\sin(x)$ function, whose period is $2\pi$ , $\sin(4x)$ repeats four times faster. This means it completes one full cycle (one rise and one fall) over an interval of length $\frac{\pi}{2}$ (about $1.57$ units on the x-axis).
Key points:
- Starts at the origin $(0,0)$ .
- Reaches its maximum value ( $1$ ) at $x = \frac{\pi}{8} \approx 0.39$ .
- Crosses the x-axis again at $x = \frac{\pi}{4} \approx 0.78$ .
- Reaches its minimum value ( $-1$ ) at $x = \frac{3\pi}{8} \approx 1.17$ .
- Completes its first cycle at $x = \frac{\pi}{2} \approx 1.57$ .
Frequency: Over one full interval of $2\pi$ , you'll see exactly 4 complete oscillations of the blue curve.

In the interactive diagram, the blue curve represents the periodic function. Red dots mark the moments when the function completes a full cycle, so you can clearly see the repetition every

\frac{2\pi}{4}

units.