Find the period and amplitude of the following function. Then sketch the function from 0 to 2π.
y = -6 sin 2π0
show your work
To find the period and amplitude of the function y = -6 sin 2π0, we can analyze the general form of the sine function, y = A sin(Bx), where A represents the amplitude and B represents the frequency.
In this case, the amplitude is given by the coefficient in front of the sine function, which is -6. Therefore, the amplitude is 6, since the negative sign only indicates a reflection across the x-axis.
The frequency, or number of cycles per unit length, is determined by B. In this case, B is 2π0. Since 2π multiplied by any number results in a full cycle, we can say that the frequency is 1 cycle per unit length. The period, which is the length of one cycle, is then given by 2π divided by the frequency, which is 2π/1 = 2π.
To sketch the function, we can consider the values of y for different values of x from 0 to 2π. Since sin(0) = 0, the y-value at x = 0 will be 0 as well. Similarly, sin(2π) = 0, so the y-value at x = 2π will also be 0.
Based on these properties, the sketch of the function y = -6 sin 2π0 from 0 to 2π will be a sinusoidal wave that starts at 0, reaches a maximum of 6, goes back to 0, reaches a minimum of -6, and ends back at 0.
The sketch of the function is a wave that starts at the origin (0, 0), rises to a peak at (π, 6), descends back to the origin at (2π, 0), continues to its lowest point at (3π, -6), and returns to the origin at (4π, 0).
Here's a rough sketch of the function y = -6 sin 2π0 from 0 to 2π:
^
6 | *
| *
| *
| *
| *
| *
| *
0 |*----------------------*-------------------->
0 π 2π 3π 4π
Remember that the sketch is based on one complete cycle, which in this case is from 0 to 2π.