Vertical Shift and Amplitude

Sine naturally swings between -1 and 1. Multiplying by a stretches that swing to between -a and a, so the wave reaches a times as high and a times as low. Adding d then takes the entire wave, swing and all, and slides it up or down by d units. Nothing about the shape changes, because you're not touching the swing anymore, just relocating where it's centered. That's why d doesn't affect amplitude: it acts after the stretching is done, like picking up the whole wave and setting it down at a new height.

The two constants answer two separate questions. Amplitude, |a|, answers 'how tall is the swing?' Vertical shift, d, answers 'where does the swing happen?' Together they build the midline, the horizontal line y = d that runs through the exact center of the wave. From the midline, the maximum is always d + |a| and the minimum is always d - |a|, since the wave reaches exactly |a| above and below center.

This splits cleanly from the horizontal transformations coming next: a and d move the wave vertically (height and position on the y-axis), while b and c inside the sine argument stretch and slide it horizontally (period and phase). Keeping these two pairs separate is what makes reading or building any sinusoid's equation manageable.