1)y=1-tan(3pi-2x) Find period and phase shift
2)y=-sin(5x+3) Find amplitude,period,and phase shift
3)y=1-sin(3pi-2x)Find amplitude,period,and phase shift
4)y=2csc(2x-1)Find period and phase shift
5)Given the graph of y = sin (x), what transformation can be made to make the graph of y = cos (x).
1) To find the period and phase shift of the function y = 1 - tan(3π - 2x), we need to analyze the equation.
The general form of tangent function is given by: y = A * tan(Bx + C) + D, where A, B, C, and D are constants.
In this case, A = 1, B = -2, C = 3π, and D = 0.
To find the period, we can use the formula: Period = 2π / |B|. In this case, |B| = |-2| = 2.
So, the period is Period = 2π / 2 = π.
To find the phase shift, we use the formula: Phase shift = -C / B. In this case, C = 3π and B = -2.
So, the phase shift is Phase shift = - (3π) / (-2) = 3π/2.
Therefore, the period of the function is π and the phase shift is 3π/2.
2) To find the amplitude, period, and phase shift of the function y = -sin(5x + 3), we analyze the equation:
The general form of sine function is given by: y = A * sin(Bx + C) + D, where A, B, C, and D are constants.
In this case, A = -1, B = 5, C = 3, and D = 0.
The amplitude of the sine function is the absolute value of A, so the amplitude here is Amplitude = |A| = |-1| = 1.
To find the period, we use the formula: Period = 2π / |B|. In this case, |B| = |5| = 5.
So, the period is Period = 2π / 5.
To find the phase shift, we use the formula: Phase shift = -C / B. In this case, C = 3 and B = 5.
So, the phase shift is Phase shift = -3 / 5.
Therefore, the amplitude is 1, the period is 2π / 5, and the phase shift is -3 / 5.
3) To find the amplitude, period, and phase shift of the function y = 1 - sin(3π - 2x), we analyze the equation:
In this case, A = 1, B = -2, C = 3π, and D = 1.
The amplitude of the sine function is the absolute value of A, so the amplitude here is Amplitude = |A| = |1| = 1.
To find the period, we use the formula: Period = 2π / |B|. In this case, |B| = |-2| = 2.
So, the period is Period = 2π / 2 = π.
To find the phase shift, we use the formula: Phase shift = -C / B. In this case, C = 3π and B = -2.
So, the phase shift is Phase shift = - (3π) / (-2) = 3π/2.
Therefore, the amplitude is 1, the period is π, and the phase shift is 3π/2.
4) To find the period and phase shift of the function y = 2csc(2x - 1), we need to analyze the equation.
The general form of csc function is given by: y = A * csc(Bx + C) + D, where A, B, C, and D are constants.
In this case, A = 2, B = 2, C = 1, and D = 0.
To find the period, we can use the formula: Period = 2π / |B|. In this case, |B| = |2| = 2.
So, the period is Period = 2π / 2 = π.
To find the phase shift, we use the formula: Phase shift = -C / B. In this case, C = 1 and B = 2.
So, the phase shift is Phase shift = -1 / 2.
Therefore, the period of the function is π and the phase shift is -1/2.
5) To make the graph of y = sin(x) into y = cos(x), we need to understand the transformation between the two functions.
The cosine function is obtained by shifting the graph of the sine function horizontally by π/2 units to the left.
So, the transformation to make the graph of y = sin(x) into y = cos(x) is to replace x with (x + π/2) in the equation. This will shift the graph π/2 units to the left.
Therefore, the transformation is y = sin(x) transformed into y = cos(x) by replacing x with (x + π/2).
1) To find the period and phase shift of y = 1 - tan(3π - 2x), we need to identify the values a, b, and c in the general form: y = a tan(bx + c).
In this case, a = 1, b = 2, and c = 3π.
The period of the tangent function is π/b, so the period for this equation is π/2.
To find the phase shift, we set the argument of the tangent function (the expression inside the parentheses) equal to 0 and solve for x.
3π - 2x = 0
2x = 3π
x = 3π/2
Therefore, the phase shift is 3π/2.
2) For y = -sin(5x + 3), we have to find the amplitude, period, and phase shift.
The general form is y = a sin(bx + c).
In this equation, a = -1, b = 5, and c = 3.
The amplitude of the sine function is |a|, so the amplitude is 1.
The period of the sine function is 2π/b, so the period in this case is 2π/5.
To find the phase shift, we set the argument of the sine function (the expression inside the parentheses) equal to 0 and solve for x.
5x + 3 = 0
5x = -3
x = -3/5
Therefore, the phase shift is -3/5.
3) For y = 1 - sin(3π - 2x), we need to determine the amplitude, period, and phase shift.
The general form is y = a + b sin(c + dx).
In this equation, a = 1, b = -1, c = 3π, and d = -2.
The amplitude is |b|, so the amplitude is 1.
The period is found by calculating 2π/|d|, which is 2π/2 = π.
To find the phase shift, we set the argument of the sine function (the expression inside the parentheses) equal to 0 and solve for x.
3π - 2x = 0
2x = 3π
x = 3π/2
Therefore, the phase shift is 3π/2.
4) For y = 2csc(2x - 1), we will determine the period and phase shift.
The general form is y = a csc(bx + c).
In this equation, a = 2, b = 2, and c = 1.
The period of the csc function is 2π/|b|, so the period in this case is 2π/2 = π.
To find the phase shift, we set the argument of the csc function (the expression inside the parentheses) equal to 0 and solve for x.
2x - 1 = 0
2x = 1
x = 1/2
Therefore, the phase shift is 1/2.
5) To transform y = sin(x) to y = cos(x), we need to apply a phase shift of π/2 or a quarter period shift to the right.
The transformation can be represented as y = sin(x - π/2). By shifting the graph of y = sin(x) by π/2 units to the right, we obtain y = cos(x).