To determine which function illustrates a change in amplitude, we need to identify the amplitude of each function. The amplitude of a trigonometric function like sine or cosine is given by the absolute value of the coefficient in front of the sine or cosine term.
Let's analyze each function one by one:
Function A: [tex]\( y = -2 - \cos(x - \pi) \)[/tex]
The cosine function here is [tex]\( \cos(x - \pi) \)[/tex]. The coefficient in front of the cosine term is [tex]\( -1 \)[/tex]. However, the sign does not affect the amplitude, only the absolute value does. Thus, the amplitude of the cosine function in Function A is:
[tex]\[ \text{Amplitude} = |1| = 1 \][/tex]
Function B: [tex]\( y = \tan(2x) \)[/tex]
The tangent function does not have a defined amplitude because it has vertical asymptotes and its values can increase or decrease without bound. Therefore, we say the amplitude of [tex]\( \tan(2x) \)[/tex] is undefined.
Function C: [tex]\( y = 1 + \sin(x) \)[/tex]
The sine function here is [tex]\( \sin(x) \)[/tex]. The coefficient in front of the sine term is [tex]\( 1 \)[/tex]. Therefore, the amplitude of the sine function in Function C is:
[tex]\[ \text{Amplitude} = |1| = 1 \][/tex]
Function D: [tex]\( y = 3 \cos(4x) \)[/tex]
The cosine function here is [tex]\( \cos(4x) \)[/tex]. The coefficient in front of the cosine term is [tex]\( 3 \)[/tex]. Therefore, the amplitude of the cosine function in Function D is:
[tex]\[ \text{Amplitude} = |3| = 3 \][/tex]
Now, comparing the amplitudes of all the given functions:
- Function A and Function C both have an amplitude of 1.
- Function B has an undefined amplitude.
- Function D has an amplitude of 3.
The question asks which function illustrates a change in amplitude. Among the given options, Function D illustrates a change in amplitude because its amplitude is different from the standard amplitude of 1 for sine and cosine functions.
Therefore, the function that illustrates a change in amplitude is:
D. [tex]\( y = 3 \cos(4x) \)[/tex]
