Write a sine function that has an amplitude of 4, period of 3π, and midline y = -3
For the function f(x)=a*sin(bx+c)+d:
Amplitude: |a| -> |4| -> a
Period: 2pi/|b| -> 2pi/|b| = 3pi -> b=2/3
Phase shift: -c/b -> 0
Midline: y=d -> y=-3
Therefore, your sine function would be f(x)=4sin((2/3)x)-3
since sin(kx) has period 2π/k, we need 2π/k = 3π
y = -3 + 4sin(2/3 x)
To write a sine function with the given amplitude, period, and midline, you can use the standard form of a sine function:
f(x) = A*sin(Bx + C) + D
Where:
- A is the amplitude
- B determines the period (B = 2π/period)
- C is the phase shift
- D is the midline
Given:
Amplitude (A) = 4
Period = 3π
Midline (D) = -3
Step 1: Determine the value of B.
Period = 2π/B
3π = 2π/B
B = 2π/(3π)
B = 2/3
Step 2: Write the sine function.
Using the information from the step above, we can write the sine function as:
f(x) = 4*sin((2/3)x + C) - 3
Since the midline is y = -3, the value of D is -3.
Therefore, the sine function with an amplitude of 4, period of 3π, and midline y = -3 is:
f(x) = 4*sin((2/3)x + C) - 3
To write a sine function with the given amplitude, period, and midline, you can use the general form of a sine function:
y = A*sin(Bx + C) + D
Where:
A represents the amplitude (in this case, 4)
B represents the frequency (which is determined by the period, in this case, the period is 3π, so B = 2π/period = 2π/(3π) = 2/3)
C represents the phase shift (we don't have any phase shift in this case, so C = 0)
D represents the midline (in this case, -3)
Therefore, the equation for the sine function would be:
y = 4*sin((2/3)x) - 3