Worksheet for writing equations of sine and cosine functions from given graphs.

Looking at Graph 4, I can see it's a sine wave with a higher frequency. The wave oscillates between y=-1 and y=1, so the amplitude is 1. It completes two full cycles from x=0 to x=2π, which means the period is π. At x=0, the function appears to be at its midline (y=0) and increasing, which is characteristic of a sine function. There's no vertical shift.

Now I'll summarize the information for each graph and write the equations.

For Graph 1:
- Amplitude: 2
- Period: 2π (so B = 2π/period = 1)
- Phase shift: 0
- Vertical shift: 0
- Equation: y = 2 sin(x)

For Graph 2:
- Amplitude: 1
- Period: 2π (so B = 1)
- Phase shift: 0
- Vertical shift: +3
- Equation: y = cos(x) + 3

For Graph 3:
- Amplitude: 1
- Period: 2π (so B = 1)
- Phase shift: 0
- Vertical shift: +1
- Equation: y = sin(x) + 1

For Graph 4:
- Amplitude: 1
- Period: π (so B = 2π/period = 2)
- Phase shift: 0
- Vertical shift: 0
- Equation: y = sin(2x)

Let me double-check these equations against the graphs to make sure they're correct.

Actually, let me reconsider Graph 3. Looking at it again, at x=0, the function appears to be at its maximum value, not at the midline. This would suggest it's actually a cosine function, not a sine function. Let me verify this.

If at x=0, the function is at its maximum (which appears to be y=2), and the center line is at y=1, then yes, this is a cosine function with amplitude 1 and vertical shift +1.

So for Graph 3:
- Amplitude: 1
- Period: 2π (so B = 1)
- Phase shift: 0
- Vertical shift: +1
- Equation: y = cos(x) + 1

Let me also check Graph 1 again. At x=0, the function is at y=0 and increasing, which is consistent with a sine function.

And for Graph 4, at x=0, the function is at y=0 and decreasing (going negative), which is still consistent with a sine function (since sin(2x) at x=0 is 0, and for small positive x, sin(2x) is positive, but looking at the graph more carefully, it seems to be going negative after x=0, which would mean it's actually -sin(2x) or there might be a phase shift).

Actually, looking at Graph 4 more carefully, at x=0, the function is at y=0 and appears to be decreasing (going negative). This would be consistent with y = -sin(2x) or y = sin(2x + π). Since the problem doesn't specify which form to use, I'll use y = -sin(2x) as it's simpler.

Final answers:

Graph 1: y = 2 sin(x)
Graph 2: y = cos(x) + 3
Graph 3: y = cos(x) + 1
Graph 4: y = -sin(2x)

I'll now fill in the requested information for each graph.