4. Write the equation of the cosine function with an amplitude of 1/3, a period of 2pi, and a phase shift of 3pi/2.
5. Write the equation of the sine function with an amplitude of 4, a period of pi/2, and a phase shift of -pi/4.
To write the equation of a cosine function, we use the general form:
f(x) = A * cos(B(x - C)) + D
where:
A is the amplitude,
B is the frequency (or period),
C is the phase shift, and
D is the vertical shift.
Similarly, to write the equation of a sine function, we use the general form:
f(x) = A * sin(B(x - C)) + D
Now, let's solve each question step-by-step.
4. Equation of the cosine function:
Given:
Amplitude (A) = 1/3
Period (B) = 2pi
Phase shift (C) = 3pi/2
Using the given values, we can write the equation as:
f(x) = (1/3) * cos(2pi(x - 3pi/2)) + D
We need to find the value of D to complete the equation. Since we don't have any information about the vertical shift (D), we can assume it to be 0. Thus, the final equation of the cosine function is:
f(x) = (1/3) * cos(2pi(x - 3pi/2))
5. Equation of the sine function:
Given:
Amplitude (A) = 4
Period (B) = pi/2
Phase shift (C) = -pi/4
Using the given values, we can write the equation as:
f(x) = 4 * sin(pi/2(x + pi/4)) + D
Similar to the previous question, since we do not have information about the vertical shift (D), we can assume it to be 0. Thus, the final equation of the sine function is:
f(x) = 4 * sin(pi/2(x + pi/4))
To write the equation of a cosine or sine function with given parameters, you use the following general forms:
Cosine function:
y = A * cos(B(x - C)) + D
Sine function:
y = A * sin(B(x - C)) + D
Where:
A represents the amplitude
B represents the frequency or the reciprocal of the period
C represents the phase shift
D represents the vertical shift
Now, let's solve each question step by step:
4. For the cosine function:
Given the amplitude (A) = 1/3, period (P) = 2π, and phase shift (C) = 3π/2.
Amplitude (A) = 1/3 represents the vertical stretch or compression of the curve. It is the value multiplying the cosine function.
Frequency (B) is the reciprocal of the period, so B = 1/P = 1/(2π).
Phase shift (C) = 3π/2 represents the horizontal translation of the curve.
Positive phase shifts moves the curve to the right, and negative phase shifts move it to the left.
The vertical shift (D) is not mentioned in the question, so we'll assume it to be zero.
Thus, the equation of the cosine function is:
y = (1/3) * cos[(1/(2π))(x - (3π/2))]
5. For the sine function:
Given the amplitude (A) = 4, period (P) = π/2, and phase shift (C) = -π/4.
Amplitude (A) = 4 represents the vertical stretch or compression of the curve. It is the value multiplying the sine function.
Frequency (B) is the reciprocal of the period, so B = 1/P = 1/(π/2) = 2/π.
Phase shift (C) = -π/4 represents the horizontal translation of the curve.
Positive phase shifts moves the curve to the right, and negative phase shifts move it to the left.
The vertical shift (D) is not mentioned in the question, so we'll assume it to be zero.
Thus, the equation of the sine function is:
y = 4 * sin[(2/π)(x - (-π/4))]
4. Y = (1/3) cos (x - 3pi/2)
5. Y = 4 sin (4x + pi/4)
The general form is
Y = (amplitude)*(sin or cos)(2 pi x/P - phase shift)
where P is the period.