To analyze the given sinusoidal function [tex]\(y = -2 \cos \left(\frac{1}{3}(x + 8)\right) + 1\)[/tex], we'll determine its key properties: amplitude, period, phase shift, and vertical shift.
1. Amplitude:
The amplitude of a sinusoidal function [tex]\(y = A \cos(Bx + C) + D\)[/tex] is given by the absolute value of [tex]\(A\)[/tex]. Here, [tex]\(A = -2\)[/tex], so the amplitude is:
[tex]\[
\text{Amplitude} = |A| = |-2| = 2
\][/tex]
2. Period:
The period of the function [tex]\(\cos(Bx)\)[/tex] is [tex]\(\frac{2\pi}{B}\)[/tex]. In our case, [tex]\(B\)[/tex] is the coefficient of [tex]\(x\)[/tex] in the argument of the cosine function, which is [tex]\(\frac{1}{3}\)[/tex]. Thus, the period is:
[tex]\[
\text{Period} = \frac{2\pi}{B} = \frac{2\pi}{\frac{1}{3}} = 2\pi \times 3 = 6\pi \approx 18.84955592153876
\][/tex]
3. Phase Shift:
The phase shift is calculated as [tex]\(\frac{-C}{B}\)[/tex], where [tex]\(C\)[/tex] is the constant added inside the cosine function. Here, [tex]\(C = 8\)[/tex], and [tex]\(B = \frac{1}{3}\)[/tex]. The phase shift is:
[tex]\[
\text{Phase Shift} = \frac{-C}{B} = \frac{-8}{\frac{1}{3}} = -8 \times 3 = -24
\][/tex]
4. Vertical Shift:
The vertical shift is given by [tex]\(D\)[/tex], the constant added to the entire function. Here, [tex]\(D = 1\)[/tex]. So, the vertical shift is:
[tex]\[
\text{Vertical Shift} = 1
\][/tex]
In summary, the function [tex]\(y = -2 \cos \left(\frac{1}{3}(x + 8)\right) + 1\)[/tex] has the following properties:
- Amplitude: [tex]\(2\)[/tex]
- Period: [tex]\(6\pi\)[/tex] or approximately [tex]\(18.84955592153876\)[/tex]
- Phase Shift: [tex]\(-24\)[/tex]
- Vertical Shift: [tex]\(1\)[/tex]
These properties completely describe the key features of the given sinusoidal function.
