What is the amplitude of `y = 3-6 sin^(2)x` ?

Options

A. 2
B. 3
C. 4
D. 6

Explanation

To find the amplitude of the function \( y = 3 - 6 \sin^2 x \), we can follow these steps:

### Step 1: Rewrite the function
The given function is:
\[ y = 3 - 6 \sin^2 x \]

### Step 2: Use the identity for sine
We know that \( \sin^2 x \) can be expressed in terms of cosine using the identity:
\[ \cos 2x = 1 - 2 \sin^2 x \]
From this, we can rearrange to find \( \sin^2 x \):
\[ \sin^2 x = \frac{1 - \cos 2x}{2} \]

### Step 3: Substitute the identity into the function
Substituting the expression for \( \sin^2 x \) into the function:
\[ y = 3 - 6 \left(\frac{1 - \cos 2x}{2}\right) \]
This simplifies to:
\[ y = 3 - 3(1 - \cos 2x) = 3 - 3 + 3\cos 2x = 3\cos 2x \]

### Step 4: Determine the amplitude
The amplitude of a function in the form \( y = A \cos(Bx) \) is given by the absolute value of \( A \). In our case:
\[ y = 3 \cos 2x \]
Here, \( A = 3 \).

### Step 5: Conclusion
Thus, the amplitude of the function \( y = 3 - 6 \sin^2 x \) is:
\[ \text{Amplitude} = |3| = 3 \]