y = sin 5x vs y = sin 2x: How Frequency Changes a Sine Graph

Grades 9-10

Question: y = sin(5x) y = sin(2x)

To understand the behavior of the trigonometric functions

y = \sin(5x)

and

y = \sin(2x)

, we must look at the standard form of a sine wave:

y = \sin(Bx)

The value

B

inside the sine function is the angular frequency. It controls how "stretched" or "compressed" the wave is horizontally.

Period ( $T$ ): This is the distance required for the function to complete one full cycle (from $0$ to $2\pi$ in the standard case). The formula for the period is: $T = \frac{2\pi}{|B|}$
Frequency: A higher value of $B$ results in a shorter period, meaning the wave oscillates more frequently within the same interval.

For $y = \sin(2x)$ (Blue Curve): The period is $T = \frac{2\pi}{2} = \pi \approx 3.14$ . In a span of $2\pi$ units, this function will complete exactly two full cycles.
For $y = \sin(5x)$ (Red Curve): The period is $T = \frac{2\pi}{5} = 0.4\pi \approx 1.26$ . In a span of $2\pi$ units, this function will complete exactly five full cycles.

When plotted together, you will notice:

Compression: The red curve ( $5x$ ) appears much "tighter" or more compressed than the blue curve ( $2x$ ).
Intersections: Both curves start at the origin $(0,0)$ . They intersect whenever $5x = 2x + 2k\pi$ or $5x = (\pi - 2x) + 2k\pi$ .
Amplitude: Notice that both functions have an amplitude of $1$ . The coefficient $B$ affects the horizontal scale (period) but does not change the vertical height (amplitude) of the peaks and valleys.

The following interactive diagram allows you to see how the number of cycles changes as you adjust the coefficients

B_1

and

B_2