how to graph y = 1/2 cos 2x for 0 ≤ x ≤ 2 p
To graph the function y = (1/2)cos(2x) for 0 ≤ x ≤ 2π, we can follow these steps:
1. Identify the period: For the cosine function, the period is given by 2π divided by the coefficient in front of x. In this case, the coefficient is 2, so the period is 2π/2 = π.
2. Determine the key points: Since the amplitude of the function is 1/2, the maximum value is 1/2 and the minimum value is -1/2. The key points on the graph occur at x = 0, π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π.
3. Plot the points: Substitute each key point into the equation y = (1/2)cos(2x) to find the corresponding y-values. Plot these points on the graph.
4. Draw the graph: Connect the plotted points with a smooth curve. The graph of y = (1/2)cos(2x) for 0 ≤ x ≤ 2π should oscillate between the maximum and minimum values over the period, creating a wave-like pattern.
Note: It is also helpful to label the x-axis with the values of π/4, π/2, 3π/4, π, 5π/4, 3π/2, 7π/4, and 2π to indicate the key points on the graph.