find the period and amplitude of each sine function.then sketch each function from 0 to 2pi:

1)y=-3.5 sin 5theta

To find the period and amplitude of the given sine function y = -3.5 sin 5θ, we need to understand a few key concepts.

First, the general form of a sine function looks like this: y = A sin(Bθ + C), where A represents the amplitude, B represents the frequency, and C represents the phase shift.

In our given function, y = -3.5 sin 5θ, the amplitude is the absolute value of the coefficient of the sine function, which is 3.5. Therefore, the amplitude is 3.5.

To find the period, we can use the formula: period = 2π/B. In our function, B is the coefficient of θ, which is 5. Plugging in this value, we get:

period = 2π/5

Now, let's sketch the function from 0 to 2π using the information we obtained.

1. Start by drawing the x-axis (θ) and the y-axis (y).

2. The amplitude tells us that the graph oscillates between -3.5 and +3.5 units vertically from the midline (y = 0). So, mark -3.5 and +3.5 on the y-axis.

3. We found that the period is 2π/5. This means that the graph completes one full cycle every 2π/5 units.

4. Divide the x-axis (θ) into equal parts, each with a length of 2π/5. Mark the points θ = 0, θ = 2π/5, θ = 4π/5, θ = 6π/5, θ = 8π/5, and θ = 2π.

5. Since the coefficient in front of θ is positive (5), the graph starts at its minimum value (amplitude value multiplied by -1) and progresses towards the maximum value.

6. Connect the plotted points smoothly to create the graph. It should show a sine wave that oscillates between -3.5 and +3.5, completing one full cycle every 2π/5 units.

That's it! You have found the period and amplitude of the given sine function and sketched it from 0 to 2π.