Question

How do you find the amplitude, period, and shift for f(x)=−4cos(3x−π)+1?

Answer 1

Answer:

(a) Amplitude = 4

(b) $T = \frac{2\pi}{3}$ --- Period

(c)

$C = \frac{\pi}{3}$ --- phase shift

$D =1$ --- vertical shift

Step-by-step explanation:

Given

$f(x) = -4\cos(3x - \pi) + 1$

Rewrite the function as:

$f(x) = -4\cos(3(x - \frac{\pi}{3}) + 1$

Solving (a): The amplitude

A cosine function is represented as:

$f(x) = A\cos[B(x - C)] + D$

Where:

$|A| \to Amplitude$

So, in this equation (by comparison):

$|A| = |-4|$

$|A| = 4$

The amplitude is 4

Solving (b): The period (T)

This is calculated as:

$T = \frac{2\pi}{B}$

By comparison:

$B =3$

So:

$T = \frac{2\pi}{3}$

Solving (c): The shift

The phase shift is C

The vertical shift is D

By comparison:

$C = \frac{\pi}{3}$ --- phase shift

$D =1$ --- vertical shift