Graph $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$. How do the graphs compare?

The given functions are $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$.

A function of the form:

$y=a\sin b\left(\theta -h\right)+k,y=a\cos b\left(\theta -h\right)+k\text{and}y=a\tan b\left(\theta -h\right)+k$ has vertical shift $\left(k\right)$. And, a period of $\frac{360°}{\left|b\right|}\text{or}\frac{2\pi }{\left|b\right|}$ for sine, cosecant, secant and cosine functions and a period of $\frac{180°}{\left|b\right|}\text{or}\frac{\pi }{\left|b\right|}$ for tangent and cotangent function. The phase shift for the functions is $\left(h\right)$.

The amplitude of secant, cosecant, tangent and cotangent functions is not defined.

The equation for the midline is written as, $y=k$. A midline is a new reference line when the parent graph is stretched vertically up or down and then the graph oscillates about new reference line called the midline.

The given functions are $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$.

The parent function $y=\cos \left(\theta \right)$is stretched vertically upwards 3 units up and its amplitude is compressed from 1 unit to $\frac{1}{2}$unit to get the transformed function $y=3-\frac{1}{2}\cos \left(\theta \right)$.

The parent function $y=\cos \left(\theta \right)$is stretched vertically upwards 3 units up and its amplitude is compressed from 1 unit to $\frac{1}{2}$unit. The parent function $y=\cos \left(\theta \right)$is shifted towards left with a phase shift of $-\pi$.

The graph of the functions $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$ are shown below.

From the graph of the function $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$ it can be interpreted that the graph of the functions $y=3-\frac{1}{2}\cos \left(\theta \right)\text{and}y=3+\frac{1}{2}\cos \left(\theta +\pi \right)$ overlap each other.