To determine the phase shift of the function [tex]\( g(x) = \frac{1}{3} \sin (2x - 5) + 1 \)[/tex], we follow these steps:
1. Identify the general form of the sine function which is [tex]\( a \sin(bx - c) + d \)[/tex].
2. Compare [tex]\( g(x) \)[/tex] to this general form to extract the values of [tex]\( b \)[/tex] and [tex]\( c \)[/tex]:
- In [tex]\( g(x) = \frac{1}{3} \sin (2x - 5) + 1 \)[/tex], we can see that:
- [tex]\( a = \frac{1}{3} \)[/tex]
- [tex]\( b = 2 \)[/tex]
- [tex]\( c = 5 \)[/tex]
- [tex]\( d = 1 \)[/tex]
3. The phase shift formula for the sine function is given by [tex]\( \frac{c}{b} \)[/tex].
Plugging in the values [tex]\( c = 5 \)[/tex] and [tex]\( b = 2 \)[/tex], we get:
[tex]\[ \text{Phase shift} = \frac{5}{2} \][/tex]
Thus, the phase shift of function [tex]\( g \)[/tex] is [tex]\( 2.5 \)[/tex].
