sketch the graph of the function.include two full periods y=2 sec 4x
sec 4x = 1/cos 4x
I can not graph it for you here
but note that cos 90 or pi/2 = 0
so you are going to have a bunch of vertical asymptotes
To sketch the graph of the function y = 2sec(4x), we need to understand the behavior of the secant function and then plot its points for two full periods.
The secant function (sec) is related to the cosine function (cos) by the reciprocal, meaning sec(x) = 1/cos(x). The cosine function has a range of values between -1 and 1, so the secant function will have values that become infinite whenever cos(x) is equal to 0, since division by zero is not defined.
To find the key points to plot the graph for two full periods, we need to identify the x-values where cos(x) = 0. These occur at the zeros of the cosine function, which are located at x = (2n + 1)(pi/2), where n is an integer.
To sketch two full periods, let's start with the first period:
1. Determine the x-values for the first period. The first period consists of the interval between two consecutive zeros of cos(x). The first zero occurs at x = (2n + 1)(pi/2), where n = 0, so the first zero is x = pi/2.
2. Find the x-values for the second zero. The second zero occurs at x = (2n + 1)(pi/2), where n = 1, so the second zero is x = (2)(pi/2) = pi.
3. Plot the important points on the graph. In the first period, we have x = pi/2 and x = pi as zeros. Calculate the corresponding y-values by evaluating y = 2sec(4x) for those x-values: y(pi/2) = 2sec(4(pi/2)) = 2sec(2pi) = 2(1) = 2, and y(pi) = 2sec(4(pi)) = 2sec(4pi) = 2(1) = 2.
4. For the second period, increase the x-values from the first period by 2pi, since it is a full period. So, the x-values for the second period will be x = pi/2 + 2pi and x = pi + 2pi = 3pi. Calculate the corresponding y-values as before: y(pi/2 + 2pi) = y(5pi/2) = 2, and y(pi + 2pi) = y(7pi) = 2.
Based on these points, we can draw the graph of y = 2sec(4x). The graph will repeat with the same shape and pattern for any additional full periods.
Please note that the secant function has vertical asymptotes at each zero of the cosine function. In this case, the vertical asymptotes will be located at x = pi/2, pi/2 + pi, pi/2 + 2pi, and so on.