what is the amplitude, period, phase shift, and vertical shift of each equation?
1. y= A cos(B(x-3))+4
2. y=-2sin(2piX/3)
3. y=4cos(3pi(x+3))
4. y= -3cos(2(x-pi/4))+0.4
thank you so much!
To determine the amplitude, period, phase shift, and vertical shift for each equation, let's break down the properties of the trigonometric functions involved.
1. For the equation y = A cos(B(x - 3)) + 4:
- Amplitude (A): The value in front of the trigonometric function determines the amplitude. In this case, the amplitude is A.
- Period: The period of the function is determined by the coefficient of x inside the trigonometric function, which is B. The period is calculated as 2π/B.
- Phase Shift: The phase shift is the horizontal translation of the graph. It occurs when (B(x - h)) is present inside the trigonometric function. In this case, the phase shift is 3 units to the right (positive) since x - 3 represents the horizontal shift.
- Vertical Shift: The vertical shift refers to the up or down translation of the graph. In this case, the graph is shifted up by 4 units.
2. For the equation y = -2sin(2πx/3):
- Amplitude: The amplitude here is the value in front of the trigonometric function, which is -2.
- Period: The period is determined by the coefficient of x inside the sine function, which is 2π/3.
- Phase Shift: There is no phase shift in this equation as there is no term inside the sine function affecting the horizontal translation.
- Vertical Shift: There is no vertical shift in this equation as there is no constant term added or subtracted.
3. For the equation y = 4cos(3π(x + 3)):
- Amplitude: The amplitude is determined by the coefficient in front of the cos function, which is 4.
- Period: The period depends on the coefficient of x inside the cosine function, which is 3π. The period is calculated as 2π/(3π), which simplifies to 2/3.
- Phase Shift: There is a phase shift of 3 units to the left (negative) as (x + 3) represents the horizontal translation.
- Vertical Shift: There is no vertical shift in this equation as there is no constant term added or subtracted.
4. For the equation y = -3cos(2(x - π/4)) + 0.4:
- Amplitude: The amplitude here is the value in front of the cosine function, which is -3.
- Period: The period is determined by the coefficient of x inside the cosine function, which is 2. The period is calculated as 2π/2, which simplifies to π.
- Phase Shift: There is a phase shift of π/4 units to the right (positive) as (x - π/4) represents the horizontal translation.
- Vertical Shift: The graph is shifted up by 0.4 units.
So, the answers are as follows:
1. Amplitude = A , Period = 2π/B , Phase Shift = 3 to the right, Vertical Shift = 4 units up
2. Amplitude = -2 , Period = 2π/(2π/3) = 3 , Phase Shift = No phase shift , Vertical Shift = No vertical shift
3. Amplitude = 4 , Period = 2π/(3π) = 2/3 , Phase Shift = 3 to the left, Vertical Shift = No vertical shift
4. Amplitude = -3 , Period = 2π/2 = π , Phase Shift = π/4 to the right, Vertical Shift = 0.4 units up