Graph the function f(x)=-1/2sin2x+1 on the interval [-2pi, 2pi]
To graph the function f(x) = -1/2sin^2(x) + 1 on the interval [-2π, 2π], follow these steps:
1. Determine the key characteristics of the function:
a. Amplitude: The coefficient before the sine squared term (-1/2) determines the amplitude. In this case, the amplitude is 1/2.
b. Vertical shift: The constant term (+1) determines the vertical shift. In this case, the function is shifted upwards by 1 unit.
c. Period: The period can be found by considering the base function sin(x). The period of sin(x) is 2π. Since the argument of the sin term in our function is 2x instead of x, the period of the function f(x) = -1/2sin^2(x) + 1 is π.
2. Identify the x-values where you want the graph to start and end. In this case, we want to graph the function on the interval [-2π, 2π], so the x-values range from -2π to 2π.
3. Divide the interval [-2π, 2π] into several equal sub-intervals of π each. The number of sub-intervals you choose will affect the accuracy of the graph. The more sub-intervals you have, the smoother and more accurate the graph will be. Let's say we choose 8 sub-intervals.
4. Calculate the y-values of the function for each x-value in the sub-intervals. Plug in the x-values into the function f(x) = -1/2sin^2(x) + 1 to get the corresponding y-values. Repeat this for each sub-interval until you have a set of (x, y) points.
5. Plot the points obtained from step 4 on a graph. Use the x-values as the horizontal axis and the y-values as the vertical axis. Connect the points with a smooth curve that represents the function.
Following these steps, you should be able to graph the function f(x) = -1/2sin^2(x) + 1 on the interval [-2π, 2π].