To cut the amplitude of the sine function [tex]\( f(x) = A \sin (Bx - C) + D \)[/tex] in half, we need to focus on the coefficient [tex]\( A \)[/tex], as it directly determines the amplitude of the function. The amplitude of a sine function [tex]\( f(x) \)[/tex] is given by the absolute value of [tex]\( A \)[/tex].
Let's outline the steps involved in modifying the equation to achieve the desired amplitude reduction:
1. Identify the initial amplitude:
The initial amplitude of the function is [tex]\( A \)[/tex].
2. Calculate the new amplitude:
To cut the amplitude in half, we simply divide the initial amplitude [tex]\( A \)[/tex] by 2.
[tex]\[
\text{New Amplitude} = \frac{A}{2}
\][/tex]
3. Modify the equation:
With the new amplitude being [tex]\( \frac{A}{2} \)[/tex], we substitute this value into the original function in place of [tex]\( A \)[/tex]. Therefore, the modified equation becomes:
[tex]\[
f(x) = \left(\frac{A}{2}\right) \sin (Bx - C) + D
\][/tex]
So, the variables in the initial equation that must be modified are specifically the amplitude [tex]\( A \)[/tex]. After the adjustment, the modified equation is:
[tex]\[
f(x) = \frac{A}{2} \sin (Bx - C) + D
\][/tex]
This modified equation ensures that the amplitude of the sine function is now half of its original value.
