Let's analyze the trigonometric equation given:
[tex]\[ y = \frac{1}{3} \cos (x + 6) \][/tex]
### Amplitude
The amplitude of a trigonometric function of the form [tex]\( y = A \cos(Bx + C) \)[/tex] or [tex]\( y = A \sin(Bx + C) \)[/tex] is the factor [tex]\( A \)[/tex] in front of the cosine or sine function.
In this equation [tex]\( y = \frac{1}{3} \cos (x + 6) \)[/tex]:
- The amplitude [tex]\( A \)[/tex] is [tex]\(\frac{1}{3}\)[/tex].
### Period
The period of a cosine function [tex]\( y = \cos(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] in front of [tex]\( x \)[/tex]. The formula for the period [tex]\( T \)[/tex] is:
[tex]\[ T = \frac{2\pi}{B} \][/tex]
In this equation, [tex]\( y = \frac{1}{3} \cos (x + 6) \)[/tex]:
- The coefficient [tex]\( B \)[/tex] of [tex]\( x \)[/tex] is 1.
- Therefore, the period [tex]\( T \)[/tex] is:
[tex]\[ T = \frac{2\pi}{1} = 2\pi \][/tex]
### Phase Shift
The phase shift of the equation [tex]\( y = A \cos(Bx + C) \)[/tex] is determined by the term [tex]\( \frac{C}{B} \)[/tex]. The phase shift direction depends on the sign of [tex]\( C \)[/tex]:
- If [tex]\( C \)[/tex] is positive, the graph is shifted to the left.
- If [tex]\( C \)[/tex] is negative, the graph is shifted to the right.
In this equation, [tex]\( y = \frac{1}{3} \cos (x + 6) \)[/tex]:
- The term inside the cosine function is [tex]\( (x + 6) \)[/tex], where [tex]\( C = 6 \)[/tex].
- The phase shift is:
[tex]\[ \frac{C}{B} = \frac{6}{1} = 6 \][/tex]
Since [tex]\( C \)[/tex] is positive, the phase shift is [tex]\( 6 \)[/tex] units to the left.
### Summary
- Amplitude: [tex]\(\frac{1}{3}\)[/tex]
- Period: [tex]\(2\pi\)[/tex]
- Phase Shift: Shifted to the left by 6 units
