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Explain which variables in the periodic function must be modified in order to cut the amplitude in half and how this would modify the equation.

The initial equation is [tex][tex]$f(x)=A \sin (B x-C)+D$[/tex][/tex].

(3 marks)



Answer :

GinnyAnswer
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.

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