To determine the amplitude, period, and phase shift of the trigonometric equation [tex]\( y = \frac{-3}{2} \cos (x) \)[/tex], let's analyze each component step-by-step.
1. Amplitude:
The amplitude of a cosine function of the form [tex]\( y = A \cos (kx + \phi) \)[/tex] is given by the absolute value of the coefficient [tex]\( A \)[/tex] before the cosine function. In this equation, the coefficient of [tex]\( \cos (x) \)[/tex] is [tex]\( -\frac{3}{2} \)[/tex]. The amplitude is therefore:
[tex]\[
\text{Amplitude} = \left| -\frac{3}{2} \right| = \frac{3}{2} = 1.5
\][/tex]
2. Period:
The period of a cosine function [tex]\( y = A \cos (kx + \phi) \)[/tex] is calculated using the formula:
[tex]\[
\text{Period} = \frac{2\pi}{k}
\][/tex]
In the given equation, the value of [tex]\( k \)[/tex] in [tex]\( \cos (x) \)[/tex] is 1. Substituting [tex]\( k = 1 \)[/tex]:
[tex]\[
\text{Period} = \frac{2\pi}{1} = 2\pi \approx 6.283185307179586
\][/tex]
3. Phase Shift:
The phase shift of a cosine function [tex]\( y = A \cos (kx + \phi) \)[/tex] is determined by the term [tex]\( \phi \)[/tex]. In the equation [tex]\( y = \frac{-3}{2} \cos (x) \)[/tex], there is no additional [tex]\( \phi \)[/tex] term (i.e., the equation is [tex]\( \cos(x + 0) \)[/tex]). This means there is no horizontal shift in the cosine function. Hence:
[tex]\[
\text{Phase Shift} = \text{no phase shift}
\][/tex]
In summary, for the equation [tex]\( y = \frac{-3}{2} \cos (x) \)[/tex]:
- Amplitude: [tex]\( 1.5 \)[/tex]
- Period: [tex]\( 6.283185307179586 \)[/tex]
- Phase Shift: no phase shift
