Sketch the graph of the function (Include two full periods. Find one complete cycle. Show your work. y=sin4x
the coefficient (4) means that the frequency is increased by a factor of four
there are four cycles in the time of one "normal" cycle
To sketch the graph of the function y = sin(4x), we can follow these steps:
Step 1: Determine the period of the function.
The period of the sine function y = sin(x) is 2π. However, when we have y = sin(4x), the argument (4x) is compressed horizontally. To find the period, we divide the original period by the value inside the sine function, in this case, 4. Thus, the period of y = sin(4x) is 2π/4, which simplifies to π/2.
Step 2: Identify key points within one complete cycle.
To sketch the graph of the function, we will focus on one complete cycle within the period. Since the period is π/2, we need to find the y-values at specific x-values within this interval.
For convenience, let's choose x = 0, π/8, π/4, π/2, 3π/8, and π/2 as our x-values.
Step 3: Calculate the corresponding y-values.
Using the x-values mentioned above, substitute them into the function y = sin(4x) to obtain their corresponding y-values.
When x = 0, y = sin(4 * 0) = sin(0) = 0
When x = π/8, y = sin(4 * (π/8)) = sin(π/2) = 1
When x = π/4, y = sin(4 * (π/4)) = sin(π) = 0
When x = π/2, y = sin(4 * (π/2)) = sin(2π) = 0
When x = 3π/8, y = sin(4 * (3π/8)) = sin(3π/2) = -1
When x = π/2, y = sin(4 * (π/2)) = sin(2π) = 0
Step 4: Plot the points and sketch the graph.
Now that we have the x and y coordinates for six points, we can plot them on a graph and connect them to sketch the graph of y = sin(4x).
The graph will start at the origin (0,0), then go up to (π/8,1), back down to (π/4,0), and remain at zero until (3π/8,-1). Finally, it returns to zero at (π/2,0).
Since we are asked to include two full periods, the graph will repeat the same pattern after one complete cycle. As a result, we can extend the graph to cover another complete cycle by mirroring the existing cycle across the y-axis.
By following these steps, you can sketch the graph of the function y = sin(4x) including two full periods.