To determine which feature of the graph of [tex]\( g(x) \)[/tex] is different from the graph of [tex]\( f(x) \)[/tex], we need to analyze each function:
1. Amplitude: The amplitude of a sinusoidal function [tex]\( A \sin(Bx - C) + D \)[/tex] is given by the absolute value of the coefficient of the sine function, [tex]\( |A| \)[/tex]. Both functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] have an amplitude of [tex]\( |5| = 5 \)[/tex]. Therefore, the amplitude is the same for both functions.
2. Vertical Shift: The vertical shift of a sinusoidal function [tex]\( A \sin(Bx - C) + D \)[/tex] is given by [tex]\( D \)[/tex]. Both functions have a vertical shift of [tex]\( -9 \)[/tex]. Therefore, the vertical shift is the same for both functions.
3. Period: The period of a sinusoidal function [tex]\( A \sin(Bx - C) + D \)[/tex] is given by [tex]\( \frac{2\pi}{|B|} \)[/tex]. For both functions, [tex]\( B = 2 \)[/tex], so the period for both is [tex]\( \frac{2\pi}{2} = \pi \)[/tex]. Therefore, the period is the same for both functions.
4. Phase Shift: The phase shift is determined by the expression [tex]\( \frac{C}{B} \)[/tex].
- For [tex]\( f(x) = 5 \sin\left(2x - \frac{\pi}{3}\right) - 9 \)[/tex], the phase shift is [tex]\( \frac{-\frac{\pi}{3}}{2} = -\frac{\pi}{6} \)[/tex]. However, without simplifying, it can be seen as [tex]\( -\frac{\pi}{3} \)[/tex].
- For [tex]\( g(x) = 5 \sin\left(2x - \frac{\pi}{8}\right) - 9 \)[/tex], the phase shift is [tex]\( \frac{-\frac{\pi}{8}}{2} = -\frac{\pi}{16} \)[/tex]. However, without simplifying, it can be seen as [tex]\( -\frac{\pi}{8} \)[/tex].
Given that for [tex]\( f(x) \)[/tex] the phase shift is [tex]\( -1.0471975511965976 \)[/tex] and for [tex]\( g(x) \)[/tex] it is [tex]\( -0.39269908169872414 \)[/tex], these numbers confirm the different phase shifts.
Thus, the feature of the graph of [tex]\( g(x) \)[/tex] that is different from the graph of [tex]\( f(x) \)[/tex] is the phase shift.
