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Find the amplitude, period, and phase shift of the function.

[tex]\[ y = -4 \cos \left(3x - \frac{\pi}{4}\right) + 3 \][/tex]

Give the exact values, not decimal approximations.

Amplitude: [tex]\(\square\)[/tex]

Period: [tex]\(\frac{2\pi}{3}\)[/tex]

Phase shift: [tex]\(\square\)[/tex]


Sagot :

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]