To analyze the trigonometric function [tex]\( y = \frac{1}{3} \cos (x + 5) \)[/tex], we need to determine three specific characteristics: the amplitude, the period, and the phase shift of the function. Let's break down each component:
1. Amplitude:
The amplitude of a cosine function [tex]\(\cos\)[/tex] is the coefficient that multiplies the function. In the given equation [tex]\( y = \frac{1}{3} \cos (x + 5) \)[/tex], the coefficient in front of the cosine function is [tex]\(\frac{1}{3}\)[/tex]. Hence, the amplitude is:
[tex]\[
\text{Amplitude} = \frac{1}{3}
\][/tex]
2. Period:
The period of a cosine function [tex]\(\cos(bx + c)\)[/tex] is determined by the coefficient [tex]\(b\)[/tex] in front of the variable [tex]\(x\)[/tex]. The general formula for the period [tex]\(T\)[/tex] is:
[tex]\[
T = \frac{2\pi}{|b|}
\][/tex]
In the given equation, the coefficient of [tex]\(x\)[/tex] is 1 (since [tex]\( \cos(x + 5) \)[/tex] can be written as [tex]\( \cos(1 \cdot x + 5) \)[/tex]). Therefore, the period [tex]\(T\)[/tex] is:
[tex]\[
T = \frac{2\pi}{1} = 2\pi
\][/tex]
3. Phase Shift:
The phase shift of a cosine function [tex]\(\cos(bx + c)\)[/tex] is found by solving for the horizontal displacement. It is given by:
[tex]\[
\text{Phase Shift} = -\frac{c}{b}
\][/tex]
In the given equation, [tex]\(c = 5\)[/tex] and [tex]\(b = 1\)[/tex], so the phase shift is:
[tex]\[
\text{Phase Shift} = -\frac{5}{1} = -5
\][/tex]
Since the phase shift is negative, it indicates a shift to the left by 5 units. Hence, the phase shift is "shifted to the left".
Summarizing the findings:
- Amplitude: [tex]\( \frac{1}{3} \)[/tex]
- Period: [tex]\( 2\pi \)[/tex]
- Phase Shift: [tex]\( -5 \)[/tex] (or "shifted to the left")
Therefore:
1. Amplitude: [tex]\(\frac{1}{3}\)[/tex]
2. Phase Shift: shifted to the left
