Question

How do you find the amplitude, period, phase shift given $y = 2 + \cos \left( {5x + \pi } \right)$?

Answer 1

Hint: We explain the main function of the given equation $y = 2 + \cos \left( {5x + \pi } \right)$ . We take the general equation and explain the amplitude, period, and shift. Then we equate the given function $y = 2 + \cos \left( {5x + \pi } \right)$ with the general one to find the value of amplitude, period, and phase shift of the equation.

Complete step-by-step solution:
We need to find the amplitude, period and shift for $y = 2 + \cos \left( {5x + \pi } \right)$.
The main function of the given equation is $\cos x$. The period of $\cos x$ is $2\pi $.
We define the general formula to explain the amplitude, period, and shift for $\cos x$.
If the $\cos x$ changes to $A\cos \left( {Bx + C} \right) + D$, the amplitude and the period becomes A and $\dfrac{{2\pi }}{B}$. The shift has two parts. One being phase shifting of the graph and the other one being vertical shift. Phase shifting is $ - \dfrac{C}{B}$ (negative sign means going left) and the vertical shift is D.
Now we explain the things for the given
$ \Rightarrow y = 2 + \cos \left( {5x + \pi } \right)$
Let's equate the equation with $A\cos \left( {Bx + C} \right) + D$.
The values will be $A = 1,B = 5,C = 1$ and $D = 2$.
So, the amplitude is 1.
The period will be,
$ \Rightarrow \dfrac{{2\pi }}{B} = \dfrac{{2\pi }}{5}$
The phase shift will be,
$ \Rightarrow - \dfrac{C}{B} = - \dfrac{\pi }{5}$

Hence, the amplitude, period, and phase shift for $y = 2 + \cos \left( {5x + \pi } \right)$ is 1, $\dfrac{{2\pi }}{5}$, and $ - \dfrac{\pi }{5}$.

Note: Amplitude is the vertical distance from the X-axis to the highest (or lowest) point on a sin or cosine curve. The period of each generalized sine or cosine curve is the length of one complete cycle. The phase shift is the amount that the curve is shifted right or left. Amplitude and period are always a positive number. The phase shift can be of both signs.