Sinusoidal Function: Amplitude, Period and Phase Shift, Interactive Graph

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Sinusoidal functions in this visualization

A sinusoidal function, or sinusoid, is a function of the form $y = A\sin(\omega x + \varphi) + k$ with $A \ne 0$ and $\omega > 0$ : the graph of $\sin x$ stretched, squeezed and shifted. Also written $A\sin\big(B(x - C)\big) + D$ , with $B = \omega$ , $C = -\varphi/\omega$ , $D = k$ . Cosine curves are sinusoids too, because $\cos x = \sin(x + \pi/2)$ . The panel turns $A$ and $\omega$ on knobs ( $\omega$ from ½ to 4) and slides the shift $-\varphi/\omega$ and $k$ on faders.

The dashed curve is the parent $y = \sin x$ ; the solid curve is the function on the panel. The yellow marks measure what the parameters set, one job each:

Four parameters, four jobs

\text{amplitude} = |A|, \quad \text{period} = \frac{2\pi}{\omega}, \quad \text{shift} = -\frac{\varphi}{\omega}, \quad \text{midline } y = k.

The vertical bracket runs from the midline to a peak (to a trough when $A < 0$ ). The horizontal bracket spans one period. The arrow along the midline goes from $x = 0$ to $x = -\varphi/\omega$ , where sin x's cycle now starts. The blue dots are the five key points of that cycle, where $\omega x + \varphi$ equals $0, \pi/2, \pi, 3\pi/2$ and $2\pi$ : the start, the extreme $y = k + A$ , the midline, the extreme $y = k - A$ and the end — a peak first and then a trough when $A > 0$ , the other way round when $A < 0$ .

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Things to notice

Inside the parentheses, changes act sideways and seem backwards: $\omega = 2$ makes the period shorter ( $\pi$ , not $4\pi$ ), and a negative $\varphi$ moves the curve to the right.
Turning $\omega$ squeezes or stretches the curve about the start of its cycle, $x = -\varphi/\omega$ : the page keeps that shift while $\omega$ turns, so $\varphi = -\omega\cdot(\text{shift})$ changes with it.
The curve swings between $k - |A|$ and $k + |A|$ , with the midline halfway. A negative $A$ flips it over the midline; the amplitude is still $|A|$ .

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Reading it correctly

The shift is $-\varphi/\omega$ , not $-\varphi$ : $\sin(2x - 2\pi/3) = \sin\big(2(x - \pi/3)\big)$ is $\sin 2x$ shifted $\pi/3$ to the right. Shifting first works too: move $\sin x$ by $-\varphi$ , then divide the x-coordinates by $\omega$ , which divides the shift as well.
The period depends only on $\omega$ , the amplitude only on $A$ . With $A = 0$ the function is the constant $k$ : the amplitude is 0, nothing oscillates and there is no least period, so the period and $\varphi$ read as dashes. $\varphi$ is computed from the shift and not reduced to any interval; $\varphi$ and $\varphi + 2n\pi$ draw the same curve.
The page keeps $\omega$ positive. A negative $\omega$ adds nothing new: since $\sin(-u) = -\sin u$ , replace $\omega$ by $|\omega|$ , $A$ by $-A$ and $\varphi$ by $-\varphi$ .

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Sketching by key points

Setting $\omega x + \varphi$ to $0, \pi/2, \pi, 3\pi/2, 2\pi$ gives $x = s,\ s + T/4,\ s + T/2,\ s + 3T/4,\ s + T$ with $s = -\varphi/\omega$ and $T = 2\pi/\omega$ , and there $y = k,\ k + A,\ k,\ k - A,\ k$ . Plotting these five points and joining them smoothly is the usual way to sketch one cycle by hand. The Key points table on the panel lists them for the function on the panel; point at a column to light its point on the graph.

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Controls

$A$ and $\omega$ on knobs; the shift $-\varphi/\omega$ on a fader that slides sideways ( $-\pi$ to $\pi$ , snapping to multiples of $\pi/12$ ) and $k$ on one that slides up and down — the way the curve moves. Click a value to type it (the shift accepts π/6, 2π/3, -pi/4 or radians). Under the shift, “≡” gives the same curve's shift the other way: shifts a whole period apart draw one curve.
On the graph, drag the start of a cycle to shift the curve, its first extreme up or down to change $A$ , and its end sideways to change the period.
Presentation mode opens a full-screen view for projectors (← → shift the curve by $\pi/12$ , ↑ ↓ move the midline by 0.5, H hides the measured values).

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