A sine function has the following key features:
Frequency = 1/4π
Amplitude = 2
Mid-line: y = 2
y-intercept: (0, 2)
The function is not a reflection of its parent function over the x-axis.
The first point must be on the mid-line and the second point must be a maximum or minimum value on the graph closest to the first point.
To find the equation of the sine function with the given key features, we can start by considering the general form of a sine function:
y = A*sin(B(x - C)) + D
where:
A is the amplitude,
B is the frequency,
C is the phase shift (horizontal shift),
D is the mid-line.
Given information:
Amplitude (A) = 2
Mid-line (D) = 2
Frequency (B) = 1/4π
Using the given information, we can substitute these values into the general form of the sine function:
y = 2*sin((1/4π)(x - C)) + 2
To find the phase shift (C), we can use the given y-intercept. The y-intercept represents when x = 0 and y = 2. Substituting these values into the equation:
2 = 2*sin((1/4π)(0 - C)) + 2
Simplifying the equation:
1 = sin(-C/4π)
To solve for C, we take the inverse sine (or arcsine) of both sides:
-C/4π = arcsin(1)
Since the sine function has a maximum value of 1, the angle whose sine is 1 is 90 degrees or π/2 radians.
-C/4π = π/2
Simplifying the equation:
-C = 2π/4
-C = π/2
To solve for C:
C = -π/2
Substituting the value of C back into the equation:
y = 2*sin((1/4π)(x + π/2)) + 2
Thus, the equation of the sine function with the given key features is:
y = 2*sin((1/4π)(x + π/2)) + 2.
To determine the equation of the given sine function, we can use the general form of a sine function:
f(x) = A * sin(Bx + C) + D
where:
A represents the amplitude,
B represents the frequency,
C represents the horizontal shift (phase shift),
D represents the vertical shift (mid-line).
Given the key features of the sine function, we have the following information:
Amplitude (A) = 2
Mid-line (D) = 2
Frequency (1 / 4π) = B
We are also told that the function is not a reflection over the x-axis, which means the B value is positive.
To find the equation using the given information, we need to determine the phase shift (C). Since the sine function starts at the mid-line (y = 2), the phase shift is 0.
Therefore, the equation of the sine function is:
f(x) = 2 * sin(1/4π * x) + 2
To verify that the first point is on the mid-line and the second point is a maximum or minimum closest to the first point, we can substitute the x-values of the points into the equation.
1. First point (0, 2):
f(0) = 2 * sin(1/4π * 0) + 2
f(0) = 2 * sin(0) + 2
f(0) = 2 * 0 + 2
f(0) = 2 + 2
f(0) = 4
The y-value at x = 0 is 4, not 2. Thus, the first point is not on the mid-line.
2. Second point (x2, y2):
To determine the x2 and y2 values, we need more information about the specific location of the second point.
Please provide additional details or coordinates to identify the second point, and I can help you further.