Let's write the general form of the sine curve:
\(f(x)=AsinB(x+C)+D\), where,
- A is the amplitude. Also, if A is negative, curve reflects about x axis
- B is the compression/stretching factor. It changes the period when it is a value other than 1.
- C is the phase shift. It translates curve left or right. Negative value shifts right and positive value shifts left.
- D is the vertical shift. It translates curve up or down. Negative value shifts down and positive value shifts up.
Let's check the 4 choices.
A.
Since this curve's A is -2, its amplitude is 2 and range is from -2 to 2. BUT since D value is -3, it shifts vertically 3 units down making the range from -1 to -5. Clearly choice A is not true.
B.
This graph is the graph of \(f(x)=-2sin(x)\) shifted 3 units DOWN since D is negative. This choice isn't true.
C.
Because of the - (minus) sign in front of the function, the function is reflected about x-axis, but amplitude doesn't change. Since A value is 2, amplitude is 2. This is true.
D.
Period depends on the B value. Here, B value is 1, so period is normal as the parent function of a sine curve, which is \(2\pi\), NOT \(4\pi\). So this is not correct.
ANSWER: C is true
