A sine function has the following key features:
Period=4pi
Amplitude: 2
Midline y=3
Y intercept; (0,3)
The function is a reflection of its parent function over the x axis.
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 is the amplitude
B is the coefficient of x that affects the period
C is a phase shift
D is the vertical shift
Given that the period is 4π, we know that B will be 2π divided by the period. So in this case, B = 2π / 4π = 1/2.
Since the function is a reflection over the x-axis, the amplitude is negative. So the amplitude, A, is -2.
The midline y=3, which means the vertical shift, D, is also 3.
Therefore, the equation of the sine function is:
f(x) = -2 * sin(1/2 * x) + 3
To determine the equation of the given sine function, we can use the standard form of the equation for a sine function:
f(x) = A*sin(Bx - C) + D
Where:
A represents the amplitude
B represents the coefficient of x which affects the period of the function
C represents the horizontal shift (phase shift) of the function
D represents the vertical shift
Given information:
Amplitude = 2
Period = 4π
Midline y = 3
Y-intercept: (0,3)
Reflection over the x-axis
Let's analyze each feature to determine the equation step-by-step.
1. Amplitude:
The amplitude is given as 2, which represents the distance from the midline to the highest or lowest point of the wave. Since the function is a reflection over the x-axis, the amplitude remains positive, so A = 2.
2. Period:
The period of a sine function is given by the formula 2π/B. In this case, the period is given as 4π. Therefore, we can set up the equation:
4π = 2π/B
Divide both sides by 2π:
2 = 1/B
Multiply both sides by B:
2B = 1
Divide both sides by 2:
B = 1/2
So, the coefficient of x, B, is 1/2.
3. Midline:
The midline is given as y = 3. The midline is the horizontal line that the wave oscillates around. For a sine function, it will be the value of D. So, D = 3.
4. Phase Shift:
Since the function is a reflection over the x-axis, there is no horizontal shift. Therefore, C = 0.
The equation of the sine function is:
f(x) = 2*sin((1/2)x) + 3
Note: The reflection over the x-axis only affects the amplitude, not the other parameters of the function.
y = 2 sin(2 pi t/T) + 3
= 2 sin(t/2) + 3