Y = 5sin (2x) + 4

What's the amplitude, period, phase shift, and vertical shift?

Y = 5 sin (2x) + 4

The amplitude of the sin itself is 1.

The amplitude of 5sin is 5.

Every time the "argument" of the sin increases by 2π

another period starts.

The first period goes from zero until 2x = 2π.

So the length of the period is π.

The phase shift is the value of the angle when x=0.

That's zero.

The sin itself wiggles up and down, regularly and symmetrically,

above and below the x-axis (y=0). In this equation, the whole thing

is shifted up 4 units by that + 4 on the end, so it's wiggling up and down

above and below the line y=4. The vertical shift is + 4.

If this function were an electrical signal, we would say that its

period is π seconds, its frequency is 1/π Hz, and it has an

AC amplitude of 5 riding on a DC component of 4.