Pls help me solve this I don't understand

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

Let me just explain, and you can rephrase it in "complete sentences".

1. y = 3sin(2x) and y = 3cos(2x)

Periods: Both have the period of π.

The reason is that sin(x) and cos(x) have period 2π.

Therefore sin(2x) and cos(2x) have half of that.

Amplitudes: Both have the amplitude of 3.

The reason is that sin(x) and cos(x) have amplitude 1.

The coefficient multiplies that by 3.

y-minimums: Both have a y-minimum -3.

The reason is that sin(x) and cos(x) have y-minimums -1.

The coefficient multiplies that by 3.

y-maximums: Both have a y-maximum 3.

The reason is that sin(x) and cos(x) have y-maximum 1.

The coefficient multiplies that by 3.

Phase shift: They are phase-shifted with respect to each other

by a quarter of their period.

That is a general relationship between sin and cos.

2. y = 4sin(2x−π) and y = cos(3x−π/2)

Periods: The first one has a period π and the other 2π/3.

In general, sin(ax + p) has a period 2π/a.

Same thing for cos(ax + p).

Amplitudes: The first one has amplitude 4 and the second 1.

Those are their coefficients.

y-minimums: The first one has a y-minimum -4 and the second -1.

Same reason as in problem 1.

y-maximums: The first one has a y-maximum 4 and the second 1.

Same reason as in problem 1.

Phase shift: To compare the phases, it is convenient to convert both to sin:

cos(3x−π/2) = sin(3x−π/2−π/2) = sin(3x−π)

So we are comparing

y = 4sin(2x−π) = 4sin(2(x-π/2)) and

y = sin(3x−π) = sin(3(x-π/3))

So the first one has phase shift π/2 and the other π/3.

Note that the two functions are in phase for

x = 0, 2π, 4π, ...

The amount by which they are out of phase with respect to each other is not constant

(as in problem 1). It is proportional to the value of x.