Question 1179802: 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)
2. y=4sin(2x-π) and y=cos(3x- π over 2)
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let's analyze the given functions and compare and contrast them:
**1. y = 3sin(2x) and y = 3cos(2x)**
* **Amplitude:** Both functions have an amplitude of 3. This means the vertical distance from the midline to the maximum or minimum value is 3.
* **Period:** Both functions have a period of π. The period is calculated as 2π / B, where B is the coefficient of x. In both cases, B = 2, so the period is 2π / 2 = π.
* **Y-Minimum:** Both functions have a y-minimum of -3.
* **Y-Maximum:** Both functions have a y-maximum of 3.
* **Phase Shift:** The two functions are related by a phase shift. Specifically, cos(x) = sin(x + π/2). Therefore:
* 3cos(2x) = 3sin(2x + π/2) = 3sin(2(x + π/4)).
* This means the cosine function is a sine function shifted π/4 units to the left.
* **Similarities:** Both functions have the same amplitude, period, y-minimum, and y-maximum. They both oscillate between -3 and 3 with a period of π.
* **Differences:** The two graphs differ only by a horizontal shift of π/4. The sine function starts at (0, 0), while the cosine function starts at (0, 3). Essentially the cosine graph is the sine graph shifted left by pi/4.
**2. y = 4sin(2x - π) and y = cos(3x - π/2)**
* **Amplitude:**
* y = 4sin(2x - π) has an amplitude of 4.
* y = cos(3x - π/2) has an amplitude of 1.
* **Period:**
* y = 4sin(2x - π) has a period of 2π / 2 = π.
* y = cos(3x - π/2) has a period of 2π / 3.
* **Y-Minimum:**
* y = 4sin(2x - π) has a y-minimum of -4.
* y = cos(3x - π/2) has a y-minimum of -1.
* **Y-Maximum:**
* y = 4sin(2x - π) has a y-maximum of 4.
* y = cos(3x - π/2) has a y-maximum of 1.
* **Phase Shift:**
* y = 4sin(2x - π) can be rewritten as 4sin(2(x - π/2)), indicating a phase shift of π/2 units to the right.
* y = cos(3x - π/2) can be rewritten as cos(3(x - π/6)), indicating a phase shift of π/6 units to the right.
* **Similarities:** Both functions represent sinusoidal curves and oscillate between their minimum and maximum values.
* **Differences:** The functions have different amplitudes, different periods, and different phase shifts.
* The sine function has a greater amplitude (4 vs 1).
* The sine function has a period of pi while the cosine has a period of 2pi/3.
* The amount of the phase shift differs as well.
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