To find the amplitude, period, and phase shift of the function [tex]\( y = -4 \cos\left(3x - \frac{\pi}{4}\right) + 3 \)[/tex], we will analyze the function step-by-step.
1. Amplitude:
The amplitude of a cosine function in the form [tex]\( y = A \cos(Bx + C) + D \)[/tex] is given by the absolute value of the coefficient in front of the cosine function. Here, the coefficient [tex]\( A \)[/tex] is [tex]\(-4\)[/tex]. The amplitude is calculated as:
[tex]\[
\text{Amplitude} = |A| = |-4| = 4
\][/tex]
2. Period:
The period of a cosine function [tex]\( y = \cos(Bx + C) \)[/tex] is determined by the coefficient [tex]\( B \)[/tex] as follows:
[tex]\[
\text{Period} = \frac{2\pi}{|B|}
\][/tex]
In our function, [tex]\( B \)[/tex] is the coefficient of [tex]\( x \)[/tex], which is [tex]\( 3 \)[/tex]. Therefore, the period is:
[tex]\[
\text{Period} = \frac{2\pi}{3}
\][/tex]
3. Phase Shift:
The phase shift of a cosine function [tex]\( y = \cos(Bx + C) \)[/tex] is given by solving for [tex]\( x \)[/tex] inside the cosine to zero:
[tex]\[
Bx - C = 0 \implies x = \frac{C}{B}
\][/tex]
In our function, [tex]\( C \)[/tex] is [tex]\(\frac{-\pi}{4}\)[/tex] (because the function is [tex]\( 3x - \frac{\pi}{4} \)[/tex]) and [tex]\( B \)[/tex] is [tex]\( 3 \)[/tex]. Therefore, the phase shift is:
[tex]\[
\text{Phase Shift} = \frac{-C}{B} = \frac{-\left(-\frac{\pi}{4}\right)}{3} = \frac{\pi}{4 \cdot 3} = \frac{\pi}{12}
\][/tex]
Putting it all together, we have:
- Amplitude: [tex]\( 4 \)[/tex]
- Period: [tex]\( \frac{2\pi}{3} \)[/tex]
- Phase Shift: [tex]\( \frac{\pi}{12} \)[/tex]
Thus, the exact values are:
[tex]\[
\text{Amplitude: } 4
\][/tex]
[tex]\[
\text{Period: } \frac{2\pi}{3}
\][/tex]
[tex]\[
\text{Phase Shift: } \frac{\pi}{12}
\][/tex]
