Amplitude, period, and phase of a sinusoid

Parametric equations

A general sinusoid

y = A sin (\frac{2 π}{T} (x - ϕ)) + k

A is the vertical scale. The amplitude is $∣ A ∣$ , and $A < 0$ flips the graph across the center line.
T is the period, the horizontal distance from one peak to the next.
φ is the phase shift. It moves the graph along the x-axis.
k moves the center line up or down.

The base sine wave $y = sin x$ is the case $A = 1$ , $T = 2 π$ , $ϕ = 0$ , $k = 0$ .

Vertical scale A: drag the slider and watch the amplitude $∣ A ∣$ , and the flip when $A < 0$ , against the base sine comparison curve
Period T: change the period and watch the wave compress or stretch, so the window holds more or fewer peaks
Phase shift φ: move where the wave starts, and watch it slip left or right of the base sine comparison curve
Center line k: shift the whole curve up or down. The guide marks $y = k$
Compare: on: show or hide the base sine wave $y = sin x$
Guides: on: show or hide the center line, the band from $k - ∣ A ∣$ to $k + ∣ A ∣$ , the phase line, and the period bracket
The sidebar shows $A$ , $T$ , $ϕ$ , and $k$ . The figure marks the current peak. When $A = 0$ the curve collapses to the center line and there is no peak.

$∣ A ∣$ sets the amplitude, and the sign of $A$ decides the flip. $T$ only changes the horizontal scale. The two are independent.
$ϕ$ is a phase shift along the x-axis. It does not change the amplitude or the period.
The angular frequency $ω = 2 π / T$ ties the period to that horizontal scale. A larger $T$ spreads the graph out.