Amplitude and Period: y = A sin(Bx)

The general sine wave has the form $y = A\sin(Bx)$. The two parameters control completely separate things:

• $|A|$ is the amplitude — the maximum height. The wave swings between $-A$ and $+A$.
• $B$ controls the frequency (how many oscillations per $2\pi$). The period is $T = 2\pi/B$.

Pulling these apart with sliders is the fastest way to internalize what each one does.

The Formulas

For $y = A\sin(Bx)$:

$\text{Amplitude} = |A| \qquad \text{Period} = T = \dfrac{2\pi}{|B|} \qquad \text{Frequency} = \dfrac{|B|}{2\pi}$

Step-by-Step Solution

Compare three waves:

1. $y_1 = \sin x$ (the baseline)
2. $y_2 = 2\sin x$ (doubled amplitude)
3. $y_3 = \sin(2x)$ (doubled frequency)

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Step 1 — Analyze $y_1 = \sin x$.

• Amplitude: $|A| = 1$
• Period: $T = 2\pi \approx 6.28$
• Maximum value: $1$, minimum: $-1$
• Zeros at $x = 0, \pi, 2\pi, \ldots$

Step 2 — Analyze $y_2 = 2\sin x$.

• $A = 2$, $B = 1$
• Amplitude: $|A| = 2$ (doubled)
• Period: $T = 2\pi/1 = 2\pi$ (unchanged)
• Maximum value: $2$, minimum: $-2$
• Zeros: same as $y_1$ (at $x = 0, \pi, 2\pi, \ldots$)

The wave is stretched vertically — it reaches further up and down, but takes the same amount of $x$ to complete one cycle.

Step 3 — Analyze $y_3 = \sin(2x)$.

• $A = 1$, $B = 2$
• Amplitude: $|A| = 1$ (unchanged)
• Period: $T = 2\pi/2 = \pi \approx 3.14$ (halved)
• Maximum value: $1$ (unchanged)
• Zeros: $2x = 0, \pi, 2\pi, 3\pi, \ldots$, so $x = 0, \pi/2, \pi, 3\pi/2, \ldots$ (twice as many!)

The wave is compressed horizontally — it completes two full cycles in the same horizontal distance that $y_1$ takes for one.

Step 4 — Tabulate values at $x = \pi/4$ for comparison.

• $y_1(\pi/4) = \sin(\pi/4) = \dfrac{\sqrt{2}}{2} \approx 0.707$
• $y_2(\pi/4) = 2\sin(\pi/4) = \sqrt{2} \approx 1.414$ (twice as tall)
• $y_3(\pi/4) = \sin(\pi/2) = 1$ (already at the peak — twice as fast)

Step 5 — Combined transformation: $y = 2\sin(2x)$.

If you use both transformations together, the wave has amplitude 2 and period $\pi$ simultaneously: tall and fast.

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Answer:

• $y_1 = \sin x$: amplitude 1, period $2\pi$
• $y_2 = 2\sin x$: amplitude 2, period $2\pi$ (vertical stretch)
• $y_3 = \sin(2x)$: amplitude 1, period $\pi$ (horizontal compression)

The two parameters $A$ and $B$ act on completely independent aspects of the wave. Adjusting one never affects the other.

Try It

• Slide A to see the wave grow taller without changing how often it oscillates.
• Slide B to see it oscillate faster without changing height.
• The HUD shows the current amplitude, period, and frequency.