To determine the amplitude and period of the function
[tex]\[ y = -3 \cos \left(\frac{4}{3} x\right), \][/tex]
we first identify the form and characteristics of a cosine function.
### Step 1: Identify the Amplitude
The amplitude of a cosine function of the form
[tex]\[ y = A \cos(Bx) \][/tex]
is given by the absolute value of the coefficient in front of the cosine function, denoted [tex]\( A \)[/tex]. In our equation, the coefficient is [tex]\(-3\)[/tex].
Therefore, the amplitude is:
[tex]\[ \text{Amplitude} = |A| = |-3| = 3. \][/tex]
### Step 2: Identify the Period
The period of a cosine function of the form
[tex]\[ y = A cos(Bx) \][/tex]
is determined by the coefficient [tex]\( B \)[/tex] in front of the variable [tex]\( x \)[/tex]. The formula for the period [tex]\( T \)[/tex] is given by:
[tex]\[ \text{Period} = \frac{2\pi}{B}. \][/tex]
In the given function, [tex]\( B = \frac{4}{3} \)[/tex].
Thus, the period is:
[tex]\[ \text{Period} = \frac{2\pi}{\frac{4}{3}} = 2\pi \times \frac{3}{4} = \frac{3}{2} \pi. \][/tex]
### Conclusion
- The amplitude of the function is [tex]\( 3 \)[/tex].
- The period of the function is [tex]\( \frac{3\pi}{2} \)[/tex].
Therefore, we have:
Period: [tex]\( \frac{3\pi}{2} \)[/tex] \
Amplitude: [tex]\( 3 \)[/tex]
