Answer :
To determine the amplitude, period, and phase shift of the trigonometric function [tex]\( y = -8 \cos(5x) \)[/tex], let's analyze the function step by step.
### Amplitude:
The amplitude of the function is the coefficient of the cosine function. In this case, the coefficient is [tex]\(-8\)[/tex].
So, the amplitude is [tex]\(|-8| = 8\)[/tex].
### Period:
To find the period of the cosine function, we use the formula for the period of [tex]\( \cos(bx) \)[/tex], which is [tex]\( \frac{2\pi}{b} \)[/tex].
Here, [tex]\( b = 5 \)[/tex], since the function is [tex]\(-8 \cos(5x)\)[/tex].
Thus, the period is [tex]\( \frac{2\pi}{5} \)[/tex].
### Phase Shift:
The phase shift of a cosine function [tex]\( \cos(bx + c) \)[/tex] is determined by the term inside the cosine function. Specifically, it would be [tex]\( \frac{-c}{b} \)[/tex].
In this function, there is no additional term inside the cosine function (no "+ c" or "- c" term), which means there is no horizontal shift.
So, there is no phase shift.
### Summary:
- Amplitude: [tex]\( 8 \)[/tex]
- Period: [tex]\( \frac{2\pi}{5} \approx 1.2566 \)[/tex]
- Phase Shift: no phase shift
These are the step-by-step details to arrive at the final answer for the amplitude, period, and phase shift of the given trigonometric function [tex]\( y = -8 \cos(5x) \)[/tex].
### Amplitude:
The amplitude of the function is the coefficient of the cosine function. In this case, the coefficient is [tex]\(-8\)[/tex].
So, the amplitude is [tex]\(|-8| = 8\)[/tex].
### Period:
To find the period of the cosine function, we use the formula for the period of [tex]\( \cos(bx) \)[/tex], which is [tex]\( \frac{2\pi}{b} \)[/tex].
Here, [tex]\( b = 5 \)[/tex], since the function is [tex]\(-8 \cos(5x)\)[/tex].
Thus, the period is [tex]\( \frac{2\pi}{5} \)[/tex].
### Phase Shift:
The phase shift of a cosine function [tex]\( \cos(bx + c) \)[/tex] is determined by the term inside the cosine function. Specifically, it would be [tex]\( \frac{-c}{b} \)[/tex].
In this function, there is no additional term inside the cosine function (no "+ c" or "- c" term), which means there is no horizontal shift.
So, there is no phase shift.
### Summary:
- Amplitude: [tex]\( 8 \)[/tex]
- Period: [tex]\( \frac{2\pi}{5} \approx 1.2566 \)[/tex]
- Phase Shift: no phase shift
These are the step-by-step details to arrive at the final answer for the amplitude, period, and phase shift of the given trigonometric function [tex]\( y = -8 \cos(5x) \)[/tex].
