To determine which function illustrates a change in amplitude, let's analyze each one:
A. [tex]\( y = -2 - \cos(x - \pi) \)[/tex]
- The cosine function, [tex]\(\cos(x - \pi)\)[/tex], has an amplitude of 1 by default.
- The negative sign in front of the cosine function does not affect the amplitude; it simply reflects the function vertically.
- The term -2 is just a vertical shift and does not affect the amplitude.
- Therefore, the amplitude remains unchanged.
B. [tex]\( y = 3 \cos 4x \)[/tex]
- The cosine function, [tex]\(\cos 4x\)[/tex], normally has an amplitude of 1.
- However, the coefficient 3 in front of the cosine function changes the amplitude.
- Specifically, this means the new amplitude is 3.
- Thus, this function does illustrate a change in amplitude.
C. [tex]\( y = \tan 2x \)[/tex]
- The tangent function, [tex]\(\tan 2x\)[/tex], does not have a maximum amplitude since its range is [tex]\(-\infty\)[/tex] to [tex]\(\infty\)[/tex].
- Therefore, it does not illustrate a change in amplitude.
D. [tex]\( y = 1 + \sin x \)[/tex]
- The sine function, [tex]\(\sin x\)[/tex], has an amplitude of 1.
- The coefficient in front of the sine function is 1, so the amplitude remains 1.
- The term +1 is just a vertical shift and does not affect the amplitude.
- Therefore, the amplitude remains unchanged.
In conclusion, the function that illustrates a change in amplitude is:
B. [tex]\( y = 3 \cos 4x \)[/tex]
