help with my math

Question

Analyzing the functions below, compare and contrast the two functions in each problem situation. Be sure to use complete sentences in your comparison. Be sure to include a discussion of similarities and differences for the periods, amplitudes, y-minimums, y-maximums, and any phase shift between the two graphs.

1. y ≡ 3sin(2x) and y = 3cos(2x)

Compare and Contrast

 

2. y = 4sin(2x−π) and y = cos(3x−π/2)

Compare and Contrast

Answer 1

1) 3sin2x and 3cos2x have the same amplitude, with maximum value 3 and minimum value = -3. both have midline y=0. they both have the same period and frequency. period = pi. frequency = 1/pi, the inverse of the period.

the only difference is a phase shift, which makes x and y intercepts different. the y intercept for the sine function, while the y intercept for the cosine function is 1.

sin2x = cos(pi/2 -2x)

3sin2x = 3cos(pi/2-2x) = 3cos(2x-pi/2) (cosine is an even function: cos2x=cos(-2x), cos(pi/2-2x) = cos(-pi/2+2x) = cos(2x-pi/2).

shift the cosine function to the right or to the left 90 degrees (pi/2) and it's the same as the sine function.

2) translate the cosine function to an equivalent sine function (or vice versa), then compare & contrast them. As in 1)